MATH 100 - FUNDAMENTALS OF
MATHEMATICS - Calculus Question Bank - Set
2
Question 1
Find the derivative of the function f(x)=3x2−4x+ 2 using the power rule
of differentiation.
Solution:
To find the derivative of the given function f(x)=3x2−4x+ 2, we will
use the power rule of differentiation, which states that the derivative of xnwith
respect to xis nxn−1.
Given function: f(x)=3x2−4x+ 2
Now, we will find the derivative of each term separately:
1. For the term 3x2, applying the power rule, the derivative is:
d
dx (3x2)=2·3x2−1= 6x
2. For the term −4x, applying the power rule, the derivative is:
d
dx (−4x) = −4·1·x1−1=−4
3. For the term 2, the derivative is zero since it is a constant.
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is:
f′(x)=6x−4
Question 1:
Find the derivative of the function f(x)=3x2−4x+ 2 using the
power rule of differentiation.
Solution:
To find the derivative of the given function f(x)=3x2−4x+ 2,
we will use the power rule of differentiation, which states that the
derivative of xnwith respect to xis nxn−1.
Given function: f(x)=3x2−4x+ 2
Now, we will find the derivative of each term separately:
1. For the term 3x2, applying the power rule, the derivative is:
1
d
dx (3x2)=2·3x2−1= 6x
2. For the term −4x, applying the power rule, the derivative is:
d
dx (−4x) = −4·1·x1−1=−4
3. For the term 2, the derivative is zero since it is a constant.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 2
Step-by-step solution: To find the derivative of the given function
f(x)=3x4+ 2x2−5x+ 7, we will differentiate each term separately
using the power rule.
1. Differentiating 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating 2x2:
d
dx (2x2)=2·2x2−1= 4x
3. Differentiating −5x:
d
dx (−5x) = −5
4. The derivative of a constant (7 in this case) is 0.
Therefore, the derivative of the function f(x)=3x4+ 2x2−5x+ 7 is:
f′(x) = 12x3+ 4x−5
Question 2: Find the derivative of the following function: f(x) =
3x4+ 2x2−5x+ 7.
Step-by-step solution: To find the derivative of the given function
f(x)=3x4+ 2x2−5x+ 7, we will differentiate each term separately
using the power rule.
1. Differentiating 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating 2x2:
d
dx (2x2)=2·2x2−1= 4x
2
3. Differentiating −5x:
d
dx (−5x) = −5
4. The derivative of a constant (7 in this case) is 0.
Therefore, the derivative of the function f(x)=3x4+ 2x2−5x+ 7 is:
f′(x) = 12x3+ 4x−5
Question 3
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we need to differentiate each term with respect to xusing
the power rule.
Step 1: Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Step 2: Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Step 3: Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Step 4: Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Step 5: Find the derivative of the constant term 9: Since the
derivative of a constant is zero,
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Certainly! Here’s a Calculus question for Liberty University along
with its step-by-step solution in LateX code:
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
3
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we need to differentiate each term with respect to xusing
the power rule.
Step 1: Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Step 2: Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Step 3: Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Step 4: Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Step 5: Find the derivative of the constant term 9: Since the
derivative of a constant is zero,
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 4
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step solution:
Given function, f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of the function, we will differentiate each
term with respect to xusing the power rule.
f′(x) = d
dx [3x4]−d
dx [2x3] + d
dx [5x2]−d
dx [7x] + d
dx [9]
Applying the power rule:
f′(x) = 12x3−6x2+ 10x−7+0
4
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−6x2+ 10x−7
Question 4:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step solution:
Given function, f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of the function, we will differentiate each
term with respect to xusing the power rule.
f′(x) = d
dx [3x4]−d
dx [2x3] + d
dx [5x2]−d
dx [7x] + d
dx [9]
Applying the power rule:
f′(x) = 12x3−6x2+ 10x−7+0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−6x2+ 10x−7
Question 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−8x2+ 2x, we will use the power rule for differentiation.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x2:
d
dx (−8x2)=2· −8x2−1=−16x
3. Find the derivative of 2x:
d
dx (2x)=1·2x1−1= 2
Therefore, the derivative of the function f(x)=3x4−8x2+ 2xis:
f′(x) = 12x3−16x+ 2
Question 5: Find the derivative of the function f(x)=3x4−8x2+ 2x.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−8x2+ 2x, we will use the power rule for differentiation.
5
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x2:
d
dx (−8x2)=2· −8x2−1=−16x
3. Find the derivative of 2x:
d
dx (2x)=1·2x1−1= 2
Therefore, the derivative of the function f(x)=3x4−8x2+ 2xis:
f′(x) = 12x3−16x+ 2
Question 6
Find the derivative of the following function: f(x)=3x2+ 5x−2.
Solution:
To find the derivative of the function f(x)=3x2+ 5x−2, we will
apply the power rule of differentiation.
Given f(x)=3x2+ 5x−2, we differentiate each term separately.
1. Differentiate 3x2:
d
dx (3x2)=3·2x
= 6x
2. Differentiate 5x:
d
dx (5x)=5
= 5
3. Differentiate −2:
d
dx (−2) = 0
= 0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Question 6:
Find the derivative of the following function: f(x)=3x2+ 5x−2.
6
Solution:
To find the derivative of the function f(x)=3x2+ 5x−2, we will
apply the power rule of differentiation.
Given f(x)=3x2+ 5x−2, we differentiate each term separately.
1. Differentiate 3x2:
d
dx (3x2)=3·2x
= 6x
2. Differentiate 5x:
d
dx (5x)=5
= 5
3. Differentiate −2:
d
dx (−2) = 0
= 0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Question 7
“‘latex Question 7:
Find the derivative of the following function:
f(x)=3x2−2x+ 5
Solution:
To find the derivative of the function f(x)=3x2−2x+ 5, we will
use the power rule and the constant rule for differentiation.
Given function: f(x)=3x2−2x+ 5
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of −2x:
d
dx (−2x) = −2
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term 5 is: d
dx (5) = 0
7
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
“‘
Feel free to let me know if you need further assistance or another
question!Sure, here is a Calculus question along with its step-by-step
solution in LateX code:
“‘latex Question 7:
Find the derivative of the following function:
f(x)=3x2−2x+ 5
Solution:
To find the derivative of the function f(x)=3x2−2x+ 5, we will
use the power rule and the constant rule for differentiation.
Given function: f(x)=3x2−2x+ 5
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of −2x:
d
dx (−2x) = −2
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term 5 is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
“‘
Feel free to let me know if you need further assistance or another
question!
Question 8
Question 8: Find the derivative of the following function:
f(x)=3x4−2x2+ 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+5, we will differentiate each term separately using the power
rule of differentiation.
8
Given function:
f(x)=3x4−2x2+ 5
Differentiating the first term using the power rule:
d
dx (3x4)=4×3x4−1= 12x3
Differentiating the second term using the power rule:
d
dx (−2x2)=2×(−2)x2−1=−4x
Differentiating the constant term:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
This completes the solution.
Feel free to reach out if you need further assistance or more ques-
tions!Certainly! Here is a question on Calculus along with a step-by-
step solution in LateX code:
Question 8: Find the derivative of the following function:
f(x)=3x4−2x2+ 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+5, we will differentiate each term separately using the power
rule of differentiation.
Given function:
f(x)=3x4−2x2+ 5
Differentiating the first term using the power rule:
d
dx (3x4)=4×3x4−1= 12x3
Differentiating the second term using the power rule:
d
dx (−2x2)=2×(−2)x2−1=−4x
Differentiating the constant term:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
This completes the solution.
Feel free to reach out if you need further assistance or more ques-
tions!
9
Question 9
Question 9: Find the derivative of the function f(x) = x3−2x2+
5x−7.
Step-by-step Solution: To find the derivative of the function, f(x) =
x3−2x2+ 5x−7, we will use the power rule:
f′(x)=3x2−4x+ 5
Therefore, the derivative of the function f(x) = x3−2x2+ 5x−7
is f′(x)=3x2−4x+ 5.Sure, here is a Calculus question along with its
step-by-step solution in LaTeX code:
Question 9: Find the derivative of the function f(x) = x3−2x2+
5x−7.
Step-by-step Solution: To find the derivative of the function, f(x) =
x3−2x2+ 5x−7, we will use the power rule:
f′(x)=3x2−4x+ 5
Therefore, the derivative of the function f(x) = x3−2x2+ 5x−7is
f′(x)=3x2−4x+ 5.
Question 10
Find the derivative of the following function using the quotient
rule:
f(x) = x2+ 3x
x+ 1
Step-by-step Solution:
To find the derivative of f(x) = x2+3x
x+1 using the quotient rule, we
can express the function as a quotient of two functions, u(x)and v(x),
and apply the quotient rule which states that if f(x) = u(x)
v(x), then
f′(x) = u′(x)v(x)−u(x)v′(x)
(v(x))2
Now, let’s compute the derivative by following these steps:
1. Express the function as a quotient of two functions:
u(x) = x2+ 3x
v(x) = x+ 1
2. Find the derivatives of u(x)and v(x):
u′(x) = 2x+ 3
10
v′(x)=1
3. Apply the quotient rule formula:
f′(x) = (2x+ 3)(x+ 1) −(x2+ 3x)(1)
(x+ 1)2
4. Simplify the expression by expanding and combining like terms:
f′(x) = 2x2+ 2x+ 3x+ 3 −x2−3x
(x+ 1)2
f′(x) = x2+ 2x+ 3
(x+ 1)2
Therefore, the derivative of the function f(x) = x2+3x
x+1 is f′(x) =
x2+2x+3
(x+1)2.Question 10:
Find the derivative of the following function using the quotient
rule:
f(x) = x2+ 3x
x+ 1
Step-by-step Solution:
To find the derivative of f(x) = x2+3x
x+1 using the quotient rule, we
can express the function as a quotient of two functions, u(x)and v(x),
and apply the quotient rule which states that if f(x) = u(x)
v(x), then
f′(x) = u′(x)v(x)−u(x)v′(x)
(v(x))2
Now, let’s compute the derivative by following these steps:
1. Express the function as a quotient of two functions:
u(x) = x2+ 3x
v(x) = x+ 1
2. Find the derivatives of u(x)and v(x):
u′(x) = 2x+ 3
v′(x)=1
3. Apply the quotient rule formula:
f′(x) = (2x+ 3)(x+ 1) −(x2+ 3x)(1)
(x+ 1)2
4. Simplify the expression by expanding and combining like terms:
f′(x) = 2x2+ 2x+ 3x+ 3 −x2−3x
(x+ 1)2
11
f′(x) = x2+ 2x+ 3
(x+ 1)2
Therefore, the derivative of the function f(x) = x2+3x
x+1 is f′(x) =
x2+2x+3
(x+1)2.
Question 11
Question 11: Find the derivative of the function f(x)=3x2+ 2x−5.
Solution: To find the derivative of the function f(x) = 3x2+ 2x−5,
we will use the power rule for derivatives.
Step 1: Identify the terms in the function:
f(x)=3x2+ 2x−5
The terms are 3x2,2x, and −5.
Step 2: Find the derivative of each term using the power rule:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
d
dx (−5) = 0
Step 3: Add the derivatives of each term to find the derivative of
the function:
f′(x) = d
dx (3x2) + d
dx (2x) + d
dx (−5) = 6x+ 2 + 0 = 6x+ 2
Therefore, the derivative of the function f(x) = 3x2+ 2x−5is
6x+ 2.Certainly! Here’s a question and its solution on Calculus for
Liberty University in LateX code:
Question 11: Find the derivative of the function f(x)=3x2+ 2x−5.
Solution: To find the derivative of the function f(x)=3x2+ 2x−5,
we will use the power rule for derivatives.
Step 1: Identify the terms in the function:
f(x)=3x2+ 2x−5
The terms are 3x2,2x, and −5.
Step 2: Find the derivative of each term using the power rule:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
12
d
dx (−5) = 0
Step 3: Add the derivatives of each term to find the derivative of
the function:
f′(x) = d
dx (3x2) + d
dx (2x) + d
dx (−5) = 6x+ 2 + 0 = 6x+ 2
Therefore, the derivative of the function f(x) = 3x2+2x−5is 6x+2.
Question 12
Question 12: Find the derivative of the function f(x)=3x4−2x3+
5x2−7using the power rule of differentiation.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7, we will differentiate each term of the function using
the power rule of differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7
Differentiate each term separately:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7) = 0
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is:
f′(x) = 12x3−6x2+ 10x
Hence, the derivative of the function is f′(x) = 12x3−6x2+10x.Certainly!
Here’s a Calculus question along with a step-by-step solution in LateX
code:
Question 12: Find the derivative of the function f(x)=3x4−2x3+
5x2−7using the power rule of differentiation.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7, we will differentiate each term of the function using
the power rule of differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7
Differentiate each term separately:
d
dx (3x4) = 4 ·3x4−1= 12x3
13
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7) = 0
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is:
f′(x) = 12x3−6x2+ 10x
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x.
Question 13
Step-by-step solution: To find the derivative of the function f(x) =
x3+4x2−2x+5, we differentiate each term of the function with respect
to xusing the power rule.
1. Differentiate the term x3:
d
dx (x3) = 3x3−1= 3x2
2. Differentiate the term 4x2:
d
dx (4x2)=2×4x2−1= 8x
3. Differentiate the term −2x:
d
dx (−2x) = −2
4. Differentiate the constant term 5: Since the derivative of a
constant is 0, the derivative of the term 5 is 0.
Putting it all together, the derivative of the function f(x) = x3+
4x2−2x+ 5 is:
f′(x) = 3x2+ 8x−2
Question 13: Compute the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
x3+4x2−2x+5, we differentiate each term of the function with respect
to xusing the power rule.
1. Differentiate the term x3:
d
dx (x3) = 3x3−1= 3x2
14
2. Differentiate the term 4x2:
d
dx (4x2)=2×4x2−1= 8x
3. Differentiate the term −2x:
d
dx (−2x) = −2
4. Differentiate the constant term 5: Since the derivative of a
constant is 0, the derivative of the term 5 is 0.
Putting it all together, the derivative of the function f(x) = x3+
4x2−2x+ 5 is:
f′(x) = 3x2+ 8x−2
Question 14
“‘latex Question 14:
Find the derivative of the function f(x) = e2xsin(x).
Step-by-step solution:
To find the derivative of the given function f(x), we will use the
product rule.
Let u(x) = e2xand v(x) = sin(x).
Then, by the product rule, the derivative of the product f(x) =
u(x)v(x)is given by:
f′(x) = u(x)v′(x) + v(x)u′(x)
Now, we find the derivatives of u(x)and v(x):
u′(x) = d
dx (e2x)=2e2x
v′(x) = d
dx (sin(x)) = cos(x)
Substitute these values into the product rule formula:
f′(x) = e2x·cos(x) + sin(x)·2e2x
Simplifying further:
f′(x) = e2xcos(x)+2e2xsin(x)
Therefore, the derivative of the function f(x) = e2xsin(x)is f′(x) =
e2xcos(x)+2e2xsin(x). “‘
Feel free to let me know if you need any further assistance!Certainly!
Here is a Calculus question along with step-by-step solutions pre-
sented in LateX code:
15
“‘latex Question 14:
Find the derivative of the function f(x) = e2xsin(x).
Step-by-step solution:
To find the derivative of the given function f(x), we will use the
product rule.
Let u(x) = e2xand v(x) = sin(x).
Then, by the product rule, the derivative of the product f(x) =
u(x)v(x)is given by:
f′(x) = u(x)v′(x) + v(x)u′(x)
Now, we find the derivatives of u(x)and v(x):
u′(x) = d
dx (e2x)=2e2x
v′(x) = d
dx (sin(x)) = cos(x)
Substitute these values into the product rule formula:
f′(x) = e2x·cos(x) + sin(x)·2e2x
Simplifying further:
f′(x) = e2xcos(x)+2e2xsin(x)
Therefore, the derivative of the function f(x) = e2xsin(x)is f′(x) =
e2xcos(x)+2e2xsin(x). “‘
Feel free to let me know if you need any further assistance!
Question 15
Find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we will differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
16
Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Find the derivative of the constant term 9:
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 15:
Find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we will differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Find the derivative of the constant term 9:
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
17
Question 16
Question 16: Find the derivative of the function f(x) = x3−4x2+
5x−2.
Step-by-step solution: To find the derivative of f(x), we will use
the power rule, which states that the derivative of xnis nxn−1.
Given function:
f(x) = x3−4x2+ 5x−2
Now, find the derivative f′(x):
f′(x) = d
dx (x3)−d
dx (4x2) + d
dx (5x)−d
dx (2)
Using the power rule:
f′(x) = 3x2−8x+ 5 −0
Therefore, the derivative of the function f(x) = x3−4x2+ 5x−2
is f′(x)=3x2−8x+ 5.Sure, here is a Calculus question along with its
step-by-step solution in LateX code:
Question 16: Find the derivative of the function f(x) = x3−4x2+
5x−2.
Step-by-step solution: To find the derivative of f(x), we will use
the power rule, which states that the derivative of xnis nxn−1.
Given function:
f(x) = x3−4x2+ 5x−2
Now, find the derivative f′(x):
f′(x) = d
dx (x3)−d
dx (4x2) + d
dx (5x)−d
dx (2)
Using the power rule:
f′(x) = 3x2−8x+ 5 −0
Therefore, the derivative of the function f(x) = x3−4x2+ 5x−2is
f′(x) = 3x2−8x+ 5.
Question 17
Question 17: Find the derivative of the function f(x)=3x2+ 4x−5.
Solution: To find the derivative of the function f(x)=3x2+ 4x−5,
we will use the power rule of derivatives.
Step 1: Find the derivative of 3x2with respect to x.
d
dx (3x2)=2·3x2−1= 6x
18
Step 2: Find the derivative of 4xwith respect to x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5with respect
to x.d
dx (−5) = 0
Hence, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Sure! Here is a Calculus question and its step-by-step solution written
in LaTeX code for Liberty University:
Question 17: Find the derivative of the function f(x)=3x2+ 4x−5.
Solution: To find the derivative of the function f(x) = 3x2+ 4x−5,
we will use the power rule of derivatives.
Step 1: Find the derivative of 3x2with respect to x.
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4xwith respect to x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5with respect
to x.d
dx (−5) = 0
Hence, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Question 18
“‘latex Question 18:
Find the derivative of the following function f(x)=3x4−2x3+5x2−
7x+ 1.
Solution:
To find the derivative of f(x), we can use the power rule of differen-
tiation. The power rule states that if f(x) = axn, then f′(x) = naxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 1
Now, let’s find the derivative of f(x):
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (1)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
19
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+1
is f′(x) = 12x3−6x2+10x−7. “‘Sure, here is the LateX code for question
18 on Calculus for Liberty University:
“‘latex Question 18:
Find the derivative of the following function f(x)=3x4−2x3+5x2−
7x+ 1.
Solution:
To find the derivative of f(x), we can use the power rule of differen-
tiation. The power rule states that if f(x) = axn, then f′(x) = naxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 1
Now, let’s find the derivative of f(x):
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (1)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+1
is f′(x) = 12x3−6x2+ 10x−7. “‘
Question 19
Question 19: Find the absolute maximum and minimum values of
the function f(x) = x3−3x2+ 4 on the interval [0,2].
Step-by-step Solution: To find the critical points of the function,
we first need to find its derivative. The derivative of f(x)is given by:
f′(x)=3x2−6x
Next, we set the derivative equal to zero and solve for xto find
the critical points:
3x2−6x= 0
3x(x−2) = 0
Solving for x, we get x= 0 and x= 2. These are the critical points.
Next, we evaluate the function f(x)at the critical points and at
the endpoints of the interval [0,2]:
f(0) = 03−3(0)2+ 4 = 4
f(2) = 23−3(2)2+ 4 = −4
Now, we compare the values of f(x)at the critical points and
endpoints to determine the absolute maximum and minimum values:
- Absolute maximum: f(0) = 4 - Absolute minimum: f(2) = −4
Therefore, the absolute maximum value of the function on the
interval [0,2] is 4 and the absolute minimum value is -4.Sure! Here is
a question on Calculus for Liberty University along with a step-by-
step solution in LateX code:
20
Question 19: Find the absolute maximum and minimum values of
the function f(x) = x3−3x2+ 4 on the interval [0,2].
Step-by-step Solution: To find the critical points of the function,
we first need to find its derivative. The derivative of f(x)is given by:
f′(x)=3x2−6x
Next, we set the derivative equal to zero and solve for xto find
the critical points:
3x2−6x= 0
3x(x−2) = 0
Solving for x, we get x= 0 and x= 2. These are the critical points.
Next, we evaluate the function f(x)at the critical points and at
the endpoints of the interval [0,2]:
f(0) = 03−3(0)2+ 4 = 4
f(2) = 23−3(2)2+ 4 = −4
Now, we compare the values of f(x)at the critical points and
endpoints to determine the absolute maximum and minimum values:
- Absolute maximum: f(0) = 4 - Absolute minimum: f(2) = −4
Therefore, the absolute maximum value of the function on the
interval [0,2] is 4 and the absolute minimum value is -4.
Question 20
Question 20: Find the derivative of the following function: f(x) =
3x4−8x3+ 2x2+ 5
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule for
derivatives.
Given function: f(x)=3x4−8x3+ 2x2+ 5
Finding the derivative of each term: 1. f′(x) = d
dx (3x4) = 4 ·3x4−1=
12x32. f′(x) = d
dx (−8x3)=3·(−8)x3−1=−24x23. f′(x) = d
dx (2x2) =
2·2x2−1= 4x4. Since the derivative of a constant is zero, f′(x) =
d
dx (5) = 0
Therefore, the derivative of the function f(x) = 3x4−8x3+ 2x2+ 5
is: f′(x) = 12x3−24x2+ 4x
Please let me know if you need any further assistance!Sure! Here
is a Calculus question along with its step-by-step solution in LateX
code:
Question 20: Find the derivative of the following function: f(x) =
3x4−8x3+ 2x2+ 5
21
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule for
derivatives.
Given function: f(x)=3x4−8x3+ 2x2+ 5
Finding the derivative of each term: 1. f′(x) = d
dx (3x4) = 4 ·3x4−1=
12x32. f′(x) = d
dx (−8x3)=3·(−8)x3−1=−24x23. f′(x) = d
dx (2x2) =
2·2x2−1= 4x4. Since the derivative of a constant is zero, f′(x) =
d
dx (5) = 0
Therefore, the derivative of the function f(x) = 3x4−8x3+ 2x2+ 5
is: f′(x) = 12x3−24x2+ 4x
Please let me know if you need any further assistance!
Question 21
Step-by-step solution: Given function: f(x)=3x4−2x2+ 5x−7
To find the derivative of the function f(x), we differentiate each
term with respect to x.
1. Differentiate the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7: Since the derivative of a
constant is zero, we have:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−4x+ 5
Question 21: Find the derivative of the following function:
f(x)=3x4−2x2+ 5x−7
Step-by-step solution: Given function: f(x)=3x4−2x2+ 5x−7
To find the derivative of the function f(x), we differentiate each
term with respect to x.
22
1. Differentiate the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7: Since the derivative of a
constant is zero, we have:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−4x+ 5
Question 22
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Since the derivative of a constant is zero, the derivative of the
term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
Question 22: Find the derivative of the function f(x)=3x4−2x2+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+ 5, we will use the power rule for differentiation.
23
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Since the derivative of a constant is zero, the derivative of the
term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
Question 23
Question 23:
Find the derivative of the function f(x) = 3x4−2x2+ 5.
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 5, we dif-
ferentiate each term with respect to xusing the power rule:
f′(x) = d
dx (3x4)−d
dx (2x2) + d
dx (5)
f′(x) = 3 ·4x4−1−2·2x2−1+ 0
f′(x) = 12x3−4x
Therefore, the derivative of the function f(x)is f′(x) = 12x3−
4x.Sure! Here’s a Calculus question for you:
Question 23:
Find the derivative of the function f(x)=3x4−2x2+ 5.
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 5, we dif-
ferentiate each term with respect to xusing the power rule:
f′(x) = d
dx (3x4)−d
dx (2x2) + d
dx (5)
f′(x) = 3 ·4x4−1−2·2x2−1+ 0
f′(x) = 12x3−4x
Therefore, the derivative of the function f(x)is f′(x) = 12x3−4x.
24
Question 24
Step-by-step solution: To find the derivative of the given function
f(x) = 3x4−5x2+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −5x2:
d
dx (−5x2)=2·(−5)x2−1=−10x
3. Find the derivative of the constant term 2: Since the derivative
of a constant is zero, the derivative of 2is 0.
Therefore, the derivative of the function f(x)=3x4−5x2+ 2 is:
f′(x) = 12x3−10x+ 0
f′(x) = 12x3−10x
Question 24: Find the derivative of the function f(x)=3x4−5x2+ 2.
Step-by-step solution: To find the derivative of the given function
f(x) = 3x4−5x2+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −5x2:
d
dx (−5x2)=2·(−5)x2−1=−10x
3. Find the derivative of the constant term 2: Since the derivative
of a constant is zero, the derivative of 2is 0.
Therefore, the derivative of the function f(x)=3x4−5x2+ 2 is:
f′(x) = 12x3−10x+ 0
f′(x) = 12x3−10x
Question 25
Question 25: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
25
Step-by-step Solution: To find the derivative of f(x)=3x4−2x3+
5x2−7x+ 9, we differentiate each term with respect to xusing the
power rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a question on Calculus along
with its step-by-step solution in LateX code:
Question 25: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step Solution: To find the derivative of f(x)=3x4−2x3+
5x2−7x+ 9, we differentiate each term with respect to xusing the
power rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 26
Question 26: Find the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
Step-by-step solution: To find the derivative of the given function
f(x) = x3+ 4x2−2x+ 5, we need to differentiate each term separately
using the power rule for differentiation.
f′(x) = d
dx (x3) + d
dx (4x2)−d
dx (2x) + d
dx (5)
f′(x)=3x2+ 8x−2+0
f′(x)=3x2+ 8x−2
Therefore, the derivative of the function f(x) = x3+ 4x2−2x+ 5 is
f′(x) = 3x2+ 8x−2.Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 26: Find the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
26
Step-by-step solution: To find the derivative of the given function
f(x) = x3+ 4x2−2x+ 5, we need to differentiate each term separately
using the power rule for differentiation.
f′(x) = d
dx (x3) + d
dx (4x2)−d
dx (2x) + d
dx (5)
f′(x)=3x2+ 8x−2+0
f′(x)=3x2+ 8x−2
Therefore, the derivative of the function f(x) = x3+ 4x2−2x+ 5 is
f′(x) = 3x2+ 8x−2.
Question 27
Question 27: Calculate the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7x−3
Step-by-step solution: Given function: f(x)=3x4+2x3−5x2+7x−3
To find the derivative of f(x), we will differentiate each term with
respect to x.
d
dx (3x4)=4·3x4−1= 12x3
d
dx (2x3)=3·2x3−1= 6x2
d
dx (−5x2)=2· −5x2−1=−10x
d
dx (7x)=1·7=7
d
dx (−3) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+7x−3
is:
f′(x) = 12x3+ 6x2−10x+ 7
This is the final answer.
Feel free to ask if you have any questions or need further clarifica-
tion.Certainly! Here’s a Calculus question along with the step-by-step
solution in LateX code:
Question 27: Calculate the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7x−3
Step-by-step solution: Given function: f(x)=3x4+2x3−5x2+7x−3
To find the derivative of f(x), we will differentiate each term with
respect to x.
d
dx (3x4)=4·3x4−1= 12x3
d
dx (2x3)=3·2x3−1= 6x2
d
dx (−5x2)=2· −5x2−1=−10x
d
dx (7x)=1·7=7
d
dx (−3) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+7x−3
is:
f′(x) = 12x3+ 6x2−10x+ 7
27
This is the final answer.
Feel free to ask if you have any questions or need further clarifi-
cation.
Question 28
Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4.
Code: “‘ Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4. “‘Sure! Here is a Calculus question
along with step-by-step solutions in LateX code:
Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
28
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4.
Code: “‘ Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4. “‘
Question 29
Question 29: Find the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will differentiate each term with respect to xusing the
power rule.
1. Differentiate the term 3x4:
d
dx (3x4) = 3 ·4x4−1= 12x3
2. Differentiate the term 2x3:
d
dx (2x3)=2·3x3−1= 6x2
3. Differentiate the term −5x2:
d
dx (−5x2) = −5·2x2−1=−10x
29
Since the derivative of a constant is zero, the derivative of 7is 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
This is the final answer.Certainly! Here’s a Calculus question
along with its solution in LateX code:
Question 29: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will differentiate each term with respect to xusing the
power rule.
1. Differentiate the term 3x4:
d
dx (3x4) = 3 ·4x4−1= 12x3
2. Differentiate the term 2x3:
d
dx (2x3)=2·3x3−1= 6x2
3. Differentiate the term −5x2:
d
dx (−5x2) = −5·2x2−1=−10x
Since the derivative of a constant is zero, the derivative of 7is 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
This is the final answer.
Question 30
Question 30: Calculate the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7Now, differentiate each
term separately: f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−0f′(x) = 12x3−
6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.
The problem has been solved.Certainly! Here is a Calculus ques-
tion along with its step-by-step solution in LateX code:
30
d
dx (3x2)=2·3x2−1= 6x
2. For the term −4x, applying the power rule, the derivative is:
d
dx (−4x) = −4·1·x1−1=−4
3. For the term 2, the derivative is zero since it is a constant.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 2
Step-by-step solution: To find the derivative of the given function
f(x)=3x4+ 2x2−5x+ 7, we will differentiate each term separately
using the power rule.
1. Differentiating 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating 2x2:
d
dx (2x2)=2·2x2−1= 4x
3. Differentiating −5x:
d
dx (−5x) = −5
4. The derivative of a constant (7 in this case) is 0.
Therefore, the derivative of the function f(x)=3x4+ 2x2−5x+ 7 is:
f′(x) = 12x3+ 4x−5
Question 2: Find the derivative of the following function: f(x) =
3x4+ 2x2−5x+ 7.
Step-by-step solution: To find the derivative of the given function
f(x)=3x4+ 2x2−5x+ 7, we will differentiate each term separately
using the power rule.
1. Differentiating 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating 2x2:
d
dx (2x2)=2·2x2−1= 4x
2
3. Differentiating −5x:
d
dx (−5x) = −5
4. The derivative of a constant (7 in this case) is 0.
Therefore, the derivative of the function f(x)=3x4+ 2x2−5x+ 7 is:
f′(x) = 12x3+ 4x−5
Question 3
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we need to differentiate each term with respect to xusing
the power rule.
Step 1: Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Step 2: Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Step 3: Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Step 4: Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Step 5: Find the derivative of the constant term 9: Since the
derivative of a constant is zero,
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Certainly! Here’s a Calculus question for Liberty University along
with its step-by-step solution in LateX code:
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
3
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we need to differentiate each term with respect to xusing
the power rule.
Step 1: Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Step 2: Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Step 3: Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Step 4: Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Step 5: Find the derivative of the constant term 9: Since the
derivative of a constant is zero,
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 4
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step solution:
Given function, f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of the function, we will differentiate each
term with respect to xusing the power rule.
f′(x) = d
dx [3x4]−d
dx [2x3] + d
dx [5x2]−d
dx [7x] + d
dx [9]
Applying the power rule:
f′(x) = 12x3−6x2+ 10x−7+0
4
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−6x2+ 10x−7
Question 4:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step solution:
Given function, f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of the function, we will differentiate each
term with respect to xusing the power rule.
f′(x) = d
dx [3x4]−d
dx [2x3] + d
dx [5x2]−d
dx [7x] + d
dx [9]
Applying the power rule:
f′(x) = 12x3−6x2+ 10x−7+0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−6x2+ 10x−7
Question 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−8x2+ 2x, we will use the power rule for differentiation.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x2:
d
dx (−8x2)=2· −8x2−1=−16x
3. Find the derivative of 2x:
d
dx (2x)=1·2x1−1= 2
Therefore, the derivative of the function f(x)=3x4−8x2+ 2xis:
f′(x) = 12x3−16x+ 2
Question 5: Find the derivative of the function f(x)=3x4−8x2+ 2x.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−8x2+ 2x, we will use the power rule for differentiation.
5
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x2:
d
dx (−8x2)=2· −8x2−1=−16x
3. Find the derivative of 2x:
d
dx (2x)=1·2x1−1= 2
Therefore, the derivative of the function f(x)=3x4−8x2+ 2xis:
f′(x) = 12x3−16x+ 2
Question 6
Find the derivative of the following function: f(x)=3x2+ 5x−2.
Solution:
To find the derivative of the function f(x)=3x2+ 5x−2, we will
apply the power rule of differentiation.
Given f(x)=3x2+ 5x−2, we differentiate each term separately.
1. Differentiate 3x2:
d
dx (3x2)=3·2x
= 6x
2. Differentiate 5x:
d
dx (5x)=5
= 5
3. Differentiate −2:
d
dx (−2) = 0
= 0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Question 6:
Find the derivative of the following function: f(x)=3x2+ 5x−2.
6
Solution:
To find the derivative of the function f(x)=3x2+ 5x−2, we will
apply the power rule of differentiation.
Given f(x)=3x2+ 5x−2, we differentiate each term separately.
1. Differentiate 3x2:
d
dx (3x2)=3·2x
= 6x
2. Differentiate 5x:
d
dx (5x)=5
= 5
3. Differentiate −2:
d
dx (−2) = 0
= 0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Question 7
“‘latex Question 7:
Find the derivative of the following function:
f(x)=3x2−2x+ 5
Solution:
To find the derivative of the function f(x)=3x2−2x+ 5, we will
use the power rule and the constant rule for differentiation.
Given function: f(x)=3x2−2x+ 5
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of −2x:
d
dx (−2x) = −2
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term 5 is: d
dx (5) = 0
7
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
“‘
Feel free to let me know if you need further assistance or another
question!Sure, here is a Calculus question along with its step-by-step
solution in LateX code:
“‘latex Question 7:
Find the derivative of the following function:
f(x)=3x2−2x+ 5
Solution:
To find the derivative of the function f(x)=3x2−2x+ 5, we will
use the power rule and the constant rule for differentiation.
Given function: f(x)=3x2−2x+ 5
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of −2x:
d
dx (−2x) = −2
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term 5 is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
“‘
Feel free to let me know if you need further assistance or another
question!
Question 8
Question 8: Find the derivative of the following function:
f(x)=3x4−2x2+ 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+5, we will differentiate each term separately using the power
rule of differentiation.
8
Given function:
f(x)=3x4−2x2+ 5
Differentiating the first term using the power rule:
d
dx (3x4)=4×3x4−1= 12x3
Differentiating the second term using the power rule:
d
dx (−2x2)=2×(−2)x2−1=−4x
Differentiating the constant term:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
This completes the solution.
Feel free to reach out if you need further assistance or more ques-
tions!Certainly! Here is a question on Calculus along with a step-by-
step solution in LateX code:
Question 8: Find the derivative of the following function:
f(x)=3x4−2x2+ 5
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+5, we will differentiate each term separately using the power
rule of differentiation.
Given function:
f(x)=3x4−2x2+ 5
Differentiating the first term using the power rule:
d
dx (3x4)=4×3x4−1= 12x3
Differentiating the second term using the power rule:
d
dx (−2x2)=2×(−2)x2−1=−4x
Differentiating the constant term:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
This completes the solution.
Feel free to reach out if you need further assistance or more ques-
tions!
9
Question 9
Question 9: Find the derivative of the function f(x) = x3−2x2+
5x−7.
Step-by-step Solution: To find the derivative of the function, f(x) =
x3−2x2+ 5x−7, we will use the power rule:
f′(x)=3x2−4x+ 5
Therefore, the derivative of the function f(x) = x3−2x2+ 5x−7
is f′(x)=3x2−4x+ 5.Sure, here is a Calculus question along with its
step-by-step solution in LaTeX code:
Question 9: Find the derivative of the function f(x) = x3−2x2+
5x−7.
Step-by-step Solution: To find the derivative of the function, f(x) =
x3−2x2+ 5x−7, we will use the power rule:
f′(x)=3x2−4x+ 5
Therefore, the derivative of the function f(x) = x3−2x2+ 5x−7is
f′(x)=3x2−4x+ 5.
Question 10
Find the derivative of the following function using the quotient
rule:
f(x) = x2+ 3x
x+ 1
Step-by-step Solution:
To find the derivative of f(x) = x2+3x
x+1 using the quotient rule, we
can express the function as a quotient of two functions, u(x)and v(x),
and apply the quotient rule which states that if f(x) = u(x)
v(x), then
f′(x) = u′(x)v(x)−u(x)v′(x)
(v(x))2
Now, let’s compute the derivative by following these steps:
1. Express the function as a quotient of two functions:
u(x) = x2+ 3x
v(x) = x+ 1
2. Find the derivatives of u(x)and v(x):
u′(x) = 2x+ 3
10
v′(x)=1
3. Apply the quotient rule formula:
f′(x) = (2x+ 3)(x+ 1) −(x2+ 3x)(1)
(x+ 1)2
4. Simplify the expression by expanding and combining like terms:
f′(x) = 2x2+ 2x+ 3x+ 3 −x2−3x
(x+ 1)2
f′(x) = x2+ 2x+ 3
(x+ 1)2
Therefore, the derivative of the function f(x) = x2+3x
x+1 is f′(x) =
x2+2x+3
(x+1)2.Question 10:
Find the derivative of the following function using the quotient
rule:
f(x) = x2+ 3x
x+ 1
Step-by-step Solution:
To find the derivative of f(x) = x2+3x
x+1 using the quotient rule, we
can express the function as a quotient of two functions, u(x)and v(x),
and apply the quotient rule which states that if f(x) = u(x)
v(x), then
f′(x) = u′(x)v(x)−u(x)v′(x)
(v(x))2
Now, let’s compute the derivative by following these steps:
1. Express the function as a quotient of two functions:
u(x) = x2+ 3x
v(x) = x+ 1
2. Find the derivatives of u(x)and v(x):
u′(x) = 2x+ 3
v′(x)=1
3. Apply the quotient rule formula:
f′(x) = (2x+ 3)(x+ 1) −(x2+ 3x)(1)
(x+ 1)2
4. Simplify the expression by expanding and combining like terms:
f′(x) = 2x2+ 2x+ 3x+ 3 −x2−3x
(x+ 1)2
11
f′(x) = x2+ 2x+ 3
(x+ 1)2
Therefore, the derivative of the function f(x) = x2+3x
x+1 is f′(x) =
x2+2x+3
(x+1)2.
Question 11
Question 11: Find the derivative of the function f(x)=3x2+ 2x−5.
Solution: To find the derivative of the function f(x) = 3x2+ 2x−5,
we will use the power rule for derivatives.
Step 1: Identify the terms in the function:
f(x)=3x2+ 2x−5
The terms are 3x2,2x, and −5.
Step 2: Find the derivative of each term using the power rule:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
d
dx (−5) = 0
Step 3: Add the derivatives of each term to find the derivative of
the function:
f′(x) = d
dx (3x2) + d
dx (2x) + d
dx (−5) = 6x+ 2 + 0 = 6x+ 2
Therefore, the derivative of the function f(x) = 3x2+ 2x−5is
6x+ 2.Certainly! Here’s a question and its solution on Calculus for
Liberty University in LateX code:
Question 11: Find the derivative of the function f(x)=3x2+ 2x−5.
Solution: To find the derivative of the function f(x)=3x2+ 2x−5,
we will use the power rule for derivatives.
Step 1: Identify the terms in the function:
f(x)=3x2+ 2x−5
The terms are 3x2,2x, and −5.
Step 2: Find the derivative of each term using the power rule:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
12
d
dx (−5) = 0
Step 3: Add the derivatives of each term to find the derivative of
the function:
f′(x) = d
dx (3x2) + d
dx (2x) + d
dx (−5) = 6x+ 2 + 0 = 6x+ 2
Therefore, the derivative of the function f(x) = 3x2+2x−5is 6x+2.
Question 12
Question 12: Find the derivative of the function f(x)=3x4−2x3+
5x2−7using the power rule of differentiation.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7, we will differentiate each term of the function using
the power rule of differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7
Differentiate each term separately:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7) = 0
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is:
f′(x) = 12x3−6x2+ 10x
Hence, the derivative of the function is f′(x) = 12x3−6x2+10x.Certainly!
Here’s a Calculus question along with a step-by-step solution in LateX
code:
Question 12: Find the derivative of the function f(x)=3x4−2x3+
5x2−7using the power rule of differentiation.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7, we will differentiate each term of the function using
the power rule of differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7
Differentiate each term separately:
d
dx (3x4) = 4 ·3x4−1= 12x3
13
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7) = 0
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is:
f′(x) = 12x3−6x2+ 10x
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x.
Question 13
Step-by-step solution: To find the derivative of the function f(x) =
x3+4x2−2x+5, we differentiate each term of the function with respect
to xusing the power rule.
1. Differentiate the term x3:
d
dx (x3) = 3x3−1= 3x2
2. Differentiate the term 4x2:
d
dx (4x2)=2×4x2−1= 8x
3. Differentiate the term −2x:
d
dx (−2x) = −2
4. Differentiate the constant term 5: Since the derivative of a
constant is 0, the derivative of the term 5 is 0.
Putting it all together, the derivative of the function f(x) = x3+
4x2−2x+ 5 is:
f′(x) = 3x2+ 8x−2
Question 13: Compute the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
x3+4x2−2x+5, we differentiate each term of the function with respect
to xusing the power rule.
1. Differentiate the term x3:
d
dx (x3) = 3x3−1= 3x2
14
2. Differentiate the term 4x2:
d
dx (4x2)=2×4x2−1= 8x
3. Differentiate the term −2x:
d
dx (−2x) = −2
4. Differentiate the constant term 5: Since the derivative of a
constant is 0, the derivative of the term 5 is 0.
Putting it all together, the derivative of the function f(x) = x3+
4x2−2x+ 5 is:
f′(x) = 3x2+ 8x−2
Question 14
“‘latex Question 14:
Find the derivative of the function f(x) = e2xsin(x).
Step-by-step solution:
To find the derivative of the given function f(x), we will use the
product rule.
Let u(x) = e2xand v(x) = sin(x).
Then, by the product rule, the derivative of the product f(x) =
u(x)v(x)is given by:
f′(x) = u(x)v′(x) + v(x)u′(x)
Now, we find the derivatives of u(x)and v(x):
u′(x) = d
dx (e2x)=2e2x
v′(x) = d
dx (sin(x)) = cos(x)
Substitute these values into the product rule formula:
f′(x) = e2x·cos(x) + sin(x)·2e2x
Simplifying further:
f′(x) = e2xcos(x)+2e2xsin(x)
Therefore, the derivative of the function f(x) = e2xsin(x)is f′(x) =
e2xcos(x)+2e2xsin(x). “‘
Feel free to let me know if you need any further assistance!Certainly!
Here is a Calculus question along with step-by-step solutions pre-
sented in LateX code:
15
“‘latex Question 14:
Find the derivative of the function f(x) = e2xsin(x).
Step-by-step solution:
To find the derivative of the given function f(x), we will use the
product rule.
Let u(x) = e2xand v(x) = sin(x).
Then, by the product rule, the derivative of the product f(x) =
u(x)v(x)is given by:
f′(x) = u(x)v′(x) + v(x)u′(x)
Now, we find the derivatives of u(x)and v(x):
u′(x) = d
dx (e2x)=2e2x
v′(x) = d
dx (sin(x)) = cos(x)
Substitute these values into the product rule formula:
f′(x) = e2x·cos(x) + sin(x)·2e2x
Simplifying further:
f′(x) = e2xcos(x)+2e2xsin(x)
Therefore, the derivative of the function f(x) = e2xsin(x)is f′(x) =
e2xcos(x)+2e2xsin(x). “‘
Feel free to let me know if you need any further assistance!
Question 15
Find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we will differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
16
Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Find the derivative of the constant term 9:
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 15:
Find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9, we will differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
Find the derivative of the constant term 9:
d
dx (9) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
17
Question 16
Question 16: Find the derivative of the function f(x) = x3−4x2+
5x−2.
Step-by-step solution: To find the derivative of f(x), we will use
the power rule, which states that the derivative of xnis nxn−1.
Given function:
f(x) = x3−4x2+ 5x−2
Now, find the derivative f′(x):
f′(x) = d
dx (x3)−d
dx (4x2) + d
dx (5x)−d
dx (2)
Using the power rule:
f′(x) = 3x2−8x+ 5 −0
Therefore, the derivative of the function f(x) = x3−4x2+ 5x−2
is f′(x)=3x2−8x+ 5.Sure, here is a Calculus question along with its
step-by-step solution in LateX code:
Question 16: Find the derivative of the function f(x) = x3−4x2+
5x−2.
Step-by-step solution: To find the derivative of f(x), we will use
the power rule, which states that the derivative of xnis nxn−1.
Given function:
f(x) = x3−4x2+ 5x−2
Now, find the derivative f′(x):
f′(x) = d
dx (x3)−d
dx (4x2) + d
dx (5x)−d
dx (2)
Using the power rule:
f′(x) = 3x2−8x+ 5 −0
Therefore, the derivative of the function f(x) = x3−4x2+ 5x−2is
f′(x) = 3x2−8x+ 5.
Question 17
Question 17: Find the derivative of the function f(x)=3x2+ 4x−5.
Solution: To find the derivative of the function f(x)=3x2+ 4x−5,
we will use the power rule of derivatives.
Step 1: Find the derivative of 3x2with respect to x.
d
dx (3x2)=2·3x2−1= 6x
18
Step 2: Find the derivative of 4xwith respect to x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5with respect
to x.d
dx (−5) = 0
Hence, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Sure! Here is a Calculus question and its step-by-step solution written
in LaTeX code for Liberty University:
Question 17: Find the derivative of the function f(x)=3x2+ 4x−5.
Solution: To find the derivative of the function f(x) = 3x2+ 4x−5,
we will use the power rule of derivatives.
Step 1: Find the derivative of 3x2with respect to x.
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4xwith respect to x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5with respect
to x.d
dx (−5) = 0
Hence, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Question 18
“‘latex Question 18:
Find the derivative of the following function f(x)=3x4−2x3+5x2−
7x+ 1.
Solution:
To find the derivative of f(x), we can use the power rule of differen-
tiation. The power rule states that if f(x) = axn, then f′(x) = naxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 1
Now, let’s find the derivative of f(x):
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (1)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
19
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+1
is f′(x) = 12x3−6x2+10x−7. “‘Sure, here is the LateX code for question
18 on Calculus for Liberty University:
“‘latex Question 18:
Find the derivative of the following function f(x)=3x4−2x3+5x2−
7x+ 1.
Solution:
To find the derivative of f(x), we can use the power rule of differen-
tiation. The power rule states that if f(x) = axn, then f′(x) = naxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 1
Now, let’s find the derivative of f(x):
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (1)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+1
is f′(x) = 12x3−6x2+ 10x−7. “‘
Question 19
Question 19: Find the absolute maximum and minimum values of
the function f(x) = x3−3x2+ 4 on the interval [0,2].
Step-by-step Solution: To find the critical points of the function,
we first need to find its derivative. The derivative of f(x)is given by:
f′(x)=3x2−6x
Next, we set the derivative equal to zero and solve for xto find
the critical points:
3x2−6x= 0
3x(x−2) = 0
Solving for x, we get x= 0 and x= 2. These are the critical points.
Next, we evaluate the function f(x)at the critical points and at
the endpoints of the interval [0,2]:
f(0) = 03−3(0)2+ 4 = 4
f(2) = 23−3(2)2+ 4 = −4
Now, we compare the values of f(x)at the critical points and
endpoints to determine the absolute maximum and minimum values:
- Absolute maximum: f(0) = 4 - Absolute minimum: f(2) = −4
Therefore, the absolute maximum value of the function on the
interval [0,2] is 4 and the absolute minimum value is -4.Sure! Here is
a question on Calculus for Liberty University along with a step-by-
step solution in LateX code:
20
Question 19: Find the absolute maximum and minimum values of
the function f(x) = x3−3x2+ 4 on the interval [0,2].
Step-by-step Solution: To find the critical points of the function,
we first need to find its derivative. The derivative of f(x)is given by:
f′(x)=3x2−6x
Next, we set the derivative equal to zero and solve for xto find
the critical points:
3x2−6x= 0
3x(x−2) = 0
Solving for x, we get x= 0 and x= 2. These are the critical points.
Next, we evaluate the function f(x)at the critical points and at
the endpoints of the interval [0,2]:
f(0) = 03−3(0)2+ 4 = 4
f(2) = 23−3(2)2+ 4 = −4
Now, we compare the values of f(x)at the critical points and
endpoints to determine the absolute maximum and minimum values:
- Absolute maximum: f(0) = 4 - Absolute minimum: f(2) = −4
Therefore, the absolute maximum value of the function on the
interval [0,2] is 4 and the absolute minimum value is -4.
Question 20
Question 20: Find the derivative of the following function: f(x) =
3x4−8x3+ 2x2+ 5
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule for
derivatives.
Given function: f(x)=3x4−8x3+ 2x2+ 5
Finding the derivative of each term: 1. f′(x) = d
dx (3x4) = 4 ·3x4−1=
12x32. f′(x) = d
dx (−8x3)=3·(−8)x3−1=−24x23. f′(x) = d
dx (2x2) =
2·2x2−1= 4x4. Since the derivative of a constant is zero, f′(x) =
d
dx (5) = 0
Therefore, the derivative of the function f(x) = 3x4−8x3+ 2x2+ 5
is: f′(x) = 12x3−24x2+ 4x
Please let me know if you need any further assistance!Sure! Here
is a Calculus question along with its step-by-step solution in LateX
code:
Question 20: Find the derivative of the following function: f(x) =
3x4−8x3+ 2x2+ 5
21
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule for
derivatives.
Given function: f(x)=3x4−8x3+ 2x2+ 5
Finding the derivative of each term: 1. f′(x) = d
dx (3x4) = 4 ·3x4−1=
12x32. f′(x) = d
dx (−8x3)=3·(−8)x3−1=−24x23. f′(x) = d
dx (2x2) =
2·2x2−1= 4x4. Since the derivative of a constant is zero, f′(x) =
d
dx (5) = 0
Therefore, the derivative of the function f(x) = 3x4−8x3+ 2x2+ 5
is: f′(x) = 12x3−24x2+ 4x
Please let me know if you need any further assistance!
Question 21
Step-by-step solution: Given function: f(x)=3x4−2x2+ 5x−7
To find the derivative of the function f(x), we differentiate each
term with respect to x.
1. Differentiate the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7: Since the derivative of a
constant is zero, we have:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−4x+ 5
Question 21: Find the derivative of the following function:
f(x)=3x4−2x2+ 5x−7
Step-by-step solution: Given function: f(x)=3x4−2x2+ 5x−7
To find the derivative of the function f(x), we differentiate each
term with respect to x.
22
1. Differentiate the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7: Since the derivative of a
constant is zero, we have:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)is:
f′(x) = 12x3−4x+ 5
Question 22
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Since the derivative of a constant is zero, the derivative of the
term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
Question 22: Find the derivative of the function f(x)=3x4−2x2+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x2+ 5, we will use the power rule for differentiation.
23
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Since the derivative of a constant is zero, the derivative of the
term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−2x2+ 5 is:
f′(x) = 12x3−4x
Question 23
Question 23:
Find the derivative of the function f(x) = 3x4−2x2+ 5.
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 5, we dif-
ferentiate each term with respect to xusing the power rule:
f′(x) = d
dx (3x4)−d
dx (2x2) + d
dx (5)
f′(x) = 3 ·4x4−1−2·2x2−1+ 0
f′(x) = 12x3−4x
Therefore, the derivative of the function f(x)is f′(x) = 12x3−
4x.Sure! Here’s a Calculus question for you:
Question 23:
Find the derivative of the function f(x)=3x4−2x2+ 5.
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 5, we dif-
ferentiate each term with respect to xusing the power rule:
f′(x) = d
dx (3x4)−d
dx (2x2) + d
dx (5)
f′(x) = 3 ·4x4−1−2·2x2−1+ 0
f′(x) = 12x3−4x
Therefore, the derivative of the function f(x)is f′(x) = 12x3−4x.
24
Question 24
Step-by-step solution: To find the derivative of the given function
f(x) = 3x4−5x2+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −5x2:
d
dx (−5x2)=2·(−5)x2−1=−10x
3. Find the derivative of the constant term 2: Since the derivative
of a constant is zero, the derivative of 2is 0.
Therefore, the derivative of the function f(x)=3x4−5x2+ 2 is:
f′(x) = 12x3−10x+ 0
f′(x) = 12x3−10x
Question 24: Find the derivative of the function f(x)=3x4−5x2+ 2.
Step-by-step solution: To find the derivative of the given function
f(x) = 3x4−5x2+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −5x2:
d
dx (−5x2)=2·(−5)x2−1=−10x
3. Find the derivative of the constant term 2: Since the derivative
of a constant is zero, the derivative of 2is 0.
Therefore, the derivative of the function f(x)=3x4−5x2+ 2 is:
f′(x) = 12x3−10x+ 0
f′(x) = 12x3−10x
Question 25
Question 25: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
25
Step-by-step Solution: To find the derivative of f(x)=3x4−2x3+
5x2−7x+ 9, we differentiate each term with respect to xusing the
power rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a question on Calculus along
with its step-by-step solution in LateX code:
Question 25: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Step-by-step Solution: To find the derivative of f(x)=3x4−2x3+
5x2−7x+ 9, we differentiate each term with respect to xusing the
power rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 26
Question 26: Find the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
Step-by-step solution: To find the derivative of the given function
f(x) = x3+ 4x2−2x+ 5, we need to differentiate each term separately
using the power rule for differentiation.
f′(x) = d
dx (x3) + d
dx (4x2)−d
dx (2x) + d
dx (5)
f′(x)=3x2+ 8x−2+0
f′(x)=3x2+ 8x−2
Therefore, the derivative of the function f(x) = x3+ 4x2−2x+ 5 is
f′(x) = 3x2+ 8x−2.Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 26: Find the derivative of the function f(x) = x3+ 4x2−
2x+ 5.
26
Step-by-step solution: To find the derivative of the given function
f(x) = x3+ 4x2−2x+ 5, we need to differentiate each term separately
using the power rule for differentiation.
f′(x) = d
dx (x3) + d
dx (4x2)−d
dx (2x) + d
dx (5)
f′(x)=3x2+ 8x−2+0
f′(x)=3x2+ 8x−2
Therefore, the derivative of the function f(x) = x3+ 4x2−2x+ 5 is
f′(x) = 3x2+ 8x−2.
Question 27
Question 27: Calculate the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7x−3
Step-by-step solution: Given function: f(x)=3x4+2x3−5x2+7x−3
To find the derivative of f(x), we will differentiate each term with
respect to x.
d
dx (3x4)=4·3x4−1= 12x3
d
dx (2x3)=3·2x3−1= 6x2
d
dx (−5x2)=2· −5x2−1=−10x
d
dx (7x)=1·7=7
d
dx (−3) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+7x−3
is:
f′(x) = 12x3+ 6x2−10x+ 7
This is the final answer.
Feel free to ask if you have any questions or need further clarifica-
tion.Certainly! Here’s a Calculus question along with the step-by-step
solution in LateX code:
Question 27: Calculate the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7x−3
Step-by-step solution: Given function: f(x)=3x4+2x3−5x2+7x−3
To find the derivative of f(x), we will differentiate each term with
respect to x.
d
dx (3x4)=4·3x4−1= 12x3
d
dx (2x3)=3·2x3−1= 6x2
d
dx (−5x2)=2· −5x2−1=−10x
d
dx (7x)=1·7=7
d
dx (−3) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+7x−3
is:
f′(x) = 12x3+ 6x2−10x+ 7
27
This is the final answer.
Feel free to ask if you have any questions or need further clarifi-
cation.
Question 28
Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4.
Code: “‘ Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4. “‘Sure! Here is a Calculus question
along with step-by-step solutions in LateX code:
Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
28
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4.
Code: “‘ Question 28: Find the limit limx→2
x2−4
x−2.
Solution: To evaluate the limit, we can first try to simplify the
expression by factoring the numerator:
lim
x→2
x2−4
x−2= lim
x→2
(x+ 2)(x−2)
x−2
Now, the (x−2) terms cancel out, leaving:
lim
x→2(x+ 2)
Substitute x= 2 into the expression:
lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, limx→2
x2−4
x−2= 4. “‘
Question 29
Question 29: Find the derivative of the following function:
f(x)=3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will differentiate each term with respect to xusing the
power rule.
1. Differentiate the term 3x4:
d
dx (3x4) = 3 ·4x4−1= 12x3
2. Differentiate the term 2x3:
d
dx (2x3)=2·3x3−1= 6x2
3. Differentiate the term −5x2:
d
dx (−5x2) = −5·2x2−1=−10x
29
Since the derivative of a constant is zero, the derivative of 7is 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
This is the final answer.Certainly! Here’s a Calculus question
along with its solution in LateX code:
Question 29: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will differentiate each term with respect to xusing the
power rule.
1. Differentiate the term 3x4:
d
dx (3x4) = 3 ·4x4−1= 12x3
2. Differentiate the term 2x3:
d
dx (2x3)=2·3x3−1= 6x2
3. Differentiate the term −5x2:
d
dx (−5x2) = −5·2x2−1=−10x
Since the derivative of a constant is zero, the derivative of 7is 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
This is the final answer.
Question 30
Question 30: Calculate the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7Now, differentiate each
term separately: f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−0f′(x) = 12x3−
6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.
The problem has been solved.Certainly! Here is a Calculus ques-
tion along with its step-by-step solution in LateX code:
30
Question 30: Calculate the derivative of the following function:
f(x) = 3x4−2x3+ 5x2−7
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we differentiate each term with respect to x.
Given function: f(x)=3x4−2x3+ 5x2−7Now, differentiate each
term separately: f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−0f′(x) = 12x3−
6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.
The problem has been solved.
31
