MATH 100 - FUNDAMENTALS OF
MATHEMATICS - Calculus Question Bank - Set
1
Question 1
Step-by-step solution: To find the derivative of f(x), we will use the power
rule for differentiation, which states that for any real number n, the derivative
of xnis nxn−1.
Given function: f(x)=3x2−4x+ 7
1. Find the derivative of 3x2: Derivative of 3x2= 2 ×3x2−1= 6x
2. Find the derivative of −4x: Derivative of −4x=−4×1x1−1=−4
3. Since the derivative of a constant is 0, the derivative of the constant term
7 is 0.
Therefore, the derivative of f(x) = 3x2−4x+ 7 is: f′(x)=6x−4Question
1: Find the derivative of the function f(x) = 3x2−4x+ 7.
Step-by-step solution: To find the derivative of f(x), we will use the
power rule for differentiation, which states that for any real number
n, the derivative of xnis nxn−1.
Given function: f(x)=3x2−4x+ 7
1. Find the derivative of 3x2: Derivative of 3x2=2×3x2−1=6x
2. Find the derivative of −4x: Derivative of −4x=−4×1x1−1=
−4
3. Since the derivative of a constant is 0, the derivative of the
constant term 7is 0.
Therefore, the derivative of f(x)=3x2−4x+ 7 is: f′(x) = 6x−4
Question 2
Question 2: Find the derivative of the following function: f(x) =
3x2+ 5x−7
Solution: To find the derivative of the function f(x) = 3x2+ 5x−7,
we’ll use the power rule for derivatives.
Step 1: Find the derivative of the term 3x2:
f′(x)=3·(2x)2−1= 6x
1
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x)=3x2+ 5x−7.Sure! Here is a question on Calculus along
with its step-by-step solution in LateX code:
Question 2: Find the derivative of the following function: f(x) =
3x2+ 5x−7
Solution: To find the derivative of the function f(x) = 3x2+ 5x−7,
we’ll use the power rule for derivatives.
Step 1: Find the derivative of the term 3x2:
f′(x)=3·(2x)2−1= 6x
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x) = 3x2+ 5x−7.
Question 3
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
2
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x) = 6x+ 4.Sure! Here
is a question on Calculus along with step-by-step solutions in LaTeX
code:
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x)=6x+ 4.
Question 4
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
3
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7
is f′(x) = 12x2−4x+ 5.Question 4: Find the derivative of the function
f(x) = 4x3−2x2+ 5x−7.
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is
f′(x) = 12x2−4x+ 5.
Question 5
Step-by-step solution: Given function: f(x) = 3x2−4x+ 2.
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 5: Find the derivative of the function f(x)=3x2−4x+ 2.
Step-by-step solution: Given function: f(x)=3x2−4x+ 2.
4
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 6
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.Sure, here is a Calculus question along with its solution
in LateX code:
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
5
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.
Question 7
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!Certainly! Here is a calculus question along
with the step-by-step solution in LateX code:
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!
Question 8
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
6
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is
4.Question 8:
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is 4.
7
Question 9
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 9: Find the derivative of the function f(x) = 4x3−2x2+5x−7.
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
8
Question 10
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.Question 10: Compute the derivative of the function
f(x) = 3x2−4x+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.
Question 11
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
9
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!Certainly!
Here is a Calculus question along with a step-by-step solution in La-
teX code:
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!
Question 12
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.Question 12: Find the derivative of the
following function: f(x)=3x4−2x3+ 5x2−7x+ 2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
10
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.
Question 13
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 13: Find the derivative of the function f(x) = 3x4−2x3+
5x2−x+ 7.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
11
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 14
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its solution in LateX code:
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.
12
Question 15
Step-by-step solution: To find the derivative of the given function
f(x) = 3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 15: Find the derivative of the following function: f(x) =
3x2−5x+ 2.
Step-by-step solution: To find the derivative of the given function
f(x)=3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 16
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
13
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!Certainly! Here
is a calculus question along with its solution in LaTeX code:
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!
Question 17
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
14
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need ad-
ditional explanations, feel free to ask.Sure! Here is a question on
Calculus for Liberty University along with the step-by-step solutions
in LateX code:
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need addi-
tional explanations, feel free to ask.
Question 18
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
15
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 18:
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 19
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
16
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.Sure! Here is a calculus question along with its step-by-
step solution in LateX code:
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.
Question 20
Question 20: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x)=3x2−4x+ 2,
we will use the power rule for differentiation which states that the
derivative of xnis nxn−1.
Therefore, the derivative of f(x)=3x2−4x+ 2 is:
d
dx (3x2−4x+ 2) = 3(2)x2−1−4(1)x1−1+ 0
d
dx (3x2−4x+ 2) = 6x−4
17
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x)=3x2+ 5x−7.Sure! Here is a question on Calculus along
with its step-by-step solution in LateX code:
Question 2: Find the derivative of the following function: f(x) =
3x2+ 5x−7
Solution: To find the derivative of the function f(x) = 3x2+ 5x−7,
we’ll use the power rule for derivatives.
Step 1: Find the derivative of the term 3x2:
f′(x)=3·(2x)2−1= 6x
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x) = 3x2+ 5x−7.
Question 3
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
2
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x) = 6x+ 4.Sure! Here
is a question on Calculus along with step-by-step solutions in LaTeX
code:
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x)=6x+ 4.
Question 4
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
3
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7
is f′(x) = 12x2−4x+ 5.Question 4: Find the derivative of the function
f(x) = 4x3−2x2+ 5x−7.
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is
f′(x) = 12x2−4x+ 5.
Question 5
Step-by-step solution: Given function: f(x) = 3x2−4x+ 2.
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 5: Find the derivative of the function f(x)=3x2−4x+ 2.
Step-by-step solution: Given function: f(x)=3x2−4x+ 2.
4
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 6
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.Sure, here is a Calculus question along with its solution
in LateX code:
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
5
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.
Question 7
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!Certainly! Here is a calculus question along
with the step-by-step solution in LateX code:
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!
Question 8
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
6
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is
4.Question 8:
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is 4.
7
Question 9
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 9: Find the derivative of the function f(x) = 4x3−2x2+5x−7.
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
8
Question 10
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.Question 10: Compute the derivative of the function
f(x) = 3x2−4x+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.
Question 11
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
9
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!Certainly!
Here is a Calculus question along with a step-by-step solution in La-
teX code:
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!
Question 12
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.Question 12: Find the derivative of the
following function: f(x)=3x4−2x3+ 5x2−7x+ 2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
10
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.
Question 13
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 13: Find the derivative of the function f(x) = 3x4−2x3+
5x2−x+ 7.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
11
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 14
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its solution in LateX code:
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.
12
Question 15
Step-by-step solution: To find the derivative of the given function
f(x) = 3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 15: Find the derivative of the following function: f(x) =
3x2−5x+ 2.
Step-by-step solution: To find the derivative of the given function
f(x)=3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 16
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
13
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!Certainly! Here
is a calculus question along with its solution in LaTeX code:
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!
Question 17
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
14
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need ad-
ditional explanations, feel free to ask.Sure! Here is a question on
Calculus for Liberty University along with the step-by-step solutions
in LateX code:
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need addi-
tional explanations, feel free to ask.
Question 18
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
15
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 18:
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 19
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
16
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.Sure! Here is a calculus question along with its step-by-
step solution in LateX code:
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.
Question 20
Question 20: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x)=3x2−4x+ 2,
we will use the power rule for differentiation which states that the
derivative of xnis nxn−1.
Therefore, the derivative of f(x)=3x2−4x+ 2 is:
d
dx (3x2−4x+ 2) = 3(2)x2−1−4(1)x1−1+ 0
d
dx (3x2−4x+ 2) = 6x−4
17
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x)=3x2+ 5x−7.Sure! Here is a question on Calculus along
with its step-by-step solution in LateX code:
Question 2: Find the derivative of the following function: f(x) =
3x2+ 5x−7
Solution: To find the derivative of the function f(x) = 3x2+ 5x−7,
we’ll use the power rule for derivatives.
Step 1: Find the derivative of the term 3x2:
f′(x)=3·(2x)2−1= 6x
Step 2: Find the derivative of the term 5x:
f′(x)=6x+ 5
Step 3: Since the derivative of a constant is 0, the derivative of
the constant term −7is 0.
Therefore, the derivative of the function f(x)=3x2+ 5x−7is:
f′(x)=6x+ 5
This is the step-by-step solution to finding the derivative of the
function f(x) = 3x2+ 5x−7.
Question 3
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
2
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x) = 6x+ 4.Sure! Here
is a question on Calculus along with step-by-step solutions in LaTeX
code:
Question 3: Find the derivative of the function f(x) = 3x2+ 4x−5
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−5using the
power rule, we differentiate each term individually.
Step 1: Find the derivative of the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
Step 2: Find the derivative of the term 4x:
d
dx (4x)=4
Step 3: Find the derivative of the constant term −5:
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
Thus, the derivative of the function is f′(x)=6x+ 4.
Question 4
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
3
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7
is f′(x) = 12x2−4x+ 5.Question 4: Find the derivative of the function
f(x) = 4x3−2x2+ 5x−7.
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will apply the power rule for differentiation.
Given function: f(x)=4x3−2x2+ 5x−7
Taking the derivative with respect to x, we differentiate each term
separately:
f′(x) = d
dx (4x3)−d
dx (2x2) + d
dx (5x)−d
dx (7)
Applying the power rule:
f′(x) = 3 ·4x3−1−2·2x2−1+ 5 + 0
Simplifying the expression:
f′(x) = 12x2−4x+ 5
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is
f′(x) = 12x2−4x+ 5.
Question 5
Step-by-step solution: Given function: f(x) = 3x2−4x+ 2.
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 5: Find the derivative of the function f(x)=3x2−4x+ 2.
Step-by-step solution: Given function: f(x)=3x2−4x+ 2.
4
To find the derivative of f(x), we differentiate each term with re-
spect to xusing the power rule for derivatives:
1. Differentiate 3x2:
d
dx (3x2)=3·2x= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: The derivative of a constant
term is always zero.
Therefore, the derivative of the function f(x)=3x2−4x+ 2 is:
f′(x)=6x−4
Question 6
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.Sure, here is a Calculus question along with its solution
in LateX code:
Question 6: Find the derivative of the function f(x) = 3x2+ 2x−4.
Solution: To find the derivative of a function, we need to apply
the power rule for differentiation. The power rule states that for any
constant nand a function u(x) = xn, the derivative is u′(x) = n·xn−1.
Given f(x)=3x2+ 2x−4, we differentiate each term individually:
f′(x) = d
dx (3x2) + d
dx (2x)−d
dx (4)
= 3 ·d
dx (x2)+2·d
dx (x)−0
= 3 ·2x2−1+ 2 ·1
= 6x+ 2.
5
Therefore, the derivative of the function f(x) = 3x2+ 2x−4is
f′(x) = 6x+ 2.
Question 7
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!Certainly! Here is a calculus question along
with the step-by-step solution in LateX code:
Question 7: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of f(x), we will differentiate each
term separately using the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
The derivative of f(x)is given by:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)is f′(x) = 12x3−6x2+
10x−7.
Please let me know if you need further assistance with calculus
questions or solutions!
Question 8
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
6
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is
4.Question 8:
Find the limit, if it exists:
lim
x→2
x2−4
x−2
Step-by-step solution:
First, try to directly substitute x= 2 into the expression to evalu-
ate the limit:
lim
x→2
x2−4
x−2=22−4
2−2=0
0
The expression gives an indeterminate form of 0
0, which means we
can simplify the expression further using algebraic manipulation:
x2−4
x−2=(x−2)(x+ 2)
x−2=x+ 2
Now, we can directly substitute x= 2 into the simplified expression
to evaluate the limit:
lim
x→2
x2−4
x−2= lim
x→2(x+ 2) = 2 + 2 = 4
Therefore, the limit of the given expression as xapproaches 2 is 4.
7
Question 9
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 9: Find the derivative of the function f(x) = 4x3−2x2+5x−7.
Step-by-step solution: To find the derivative of the given function,
we will differentiate each term with respect to xusing the power rule.
Given function: f(x)=4x3−2x2+ 5x−7
Derivative of 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Derivative of −2x2:
d
dx (−2x2) = 2 · −2x2−1=−4x
Derivative of 5x:d
dx (5x)=5
Derivative of −7:d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
8
Question 10
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.Question 10: Compute the derivative of the function
f(x) = 3x2−4x+ 5.
Step-by-step solution: To find the derivative of the function f(x) =
3x2−4x+ 5, we will use the power rule of differentiation. The power
rule states that if f(x) = xn, then f′(x) = nxn−1.
1. Find f′(x):
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=3·d
dx (x2)−4·d
dx (x)+0
f′(x) = 3 ·2x−4·1
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.
Question 11
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
9
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!Certainly!
Here is a Calculus question along with a step-by-step solution in La-
teX code:
Question 11: Find the derivative of the following function: f(x) =
3x2+ 2x−5
Step-by-step solution: To find the derivative of the function f(x) =
3x2+ 2x−5, we will use the power rule for differentiation.
Given function: f(x)=3x2+ 2x−5
Now, differentiate each term with respect to x:
d
dx (3x2)=3·2x2−1= 6x
d
dx (2x)=2·1x1−1= 2
d
dx (−5) = 0 (derivative of a constant is zero)
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the final answer.
Feel free to reach out if you need any further assistance!
Question 12
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.Question 12: Find the derivative of the
following function: f(x)=3x4−2x3+ 5x2−7x+ 2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 2, we will apply the power rule for differentiation.
10
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking the derivative of each term: f′(x) = d
dx [3x4]−d
dx [2x3]+ d
dx [5x2]−
d
dx [7x] + d
dx [2]
Applying the power rule: f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+2
is f′(x) = 12x3−6x2+ 10x−7.
Question 13
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 13: Find the derivative of the function f(x) = 3x4−2x3+
5x2−x+ 7.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−x+ 7, we need to differentiate each term separately
using the power rule.
1. Differentiating the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
11
2. Differentiating the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiating the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiating the term −x:
d
dx (−x) = 1 ·(−1)x1−1=−1
5. Differentiating the constant term 7:
d
dx (7) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−x+7
is:
f′(x) = 12x3−6x2+ 10x−1
Question 14
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its solution in LateX code:
Question 14: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 8.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 8, we will differentiate each term with respect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (8)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1+0
= 12x3−6x2+ 10x−7.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+8
is f′(x) = 12x3−6x2+ 10x−7.
12
Question 15
Step-by-step solution: To find the derivative of the given function
f(x) = 3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 15: Find the derivative of the following function: f(x) =
3x2−5x+ 2.
Step-by-step solution: To find the derivative of the given function
f(x)=3x2−5x+ 2, we will use the power rule for derivatives.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of −5x:
d
dx (−5x) = −5
3. Find the derivative of the constant term 2: Since the derivative
of a constant is always zero, the derivative of the constant term 2is
0.
Therefore, the derivative of the function f(x)=3x2−5x+ 2 is:
f′(x)=6x−5
Question 16
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
13
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!Certainly! Here
is a calculus question along with its solution in LaTeX code:
Question 16:
Find the derivative of the following function:
f(x) = 2x3+ 5x2−4x+ 7
x2+ 3x
Solution:
To find the derivative of the given function f(x), we will use the
quotient rule:
If f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Given function: f(x) = 2x3+5x2−4x+7
x2+3x
Let g(x)=2x3+ 5x2−4x+ 7 and h(x) = x2+ 3x.
Now, find the derivatives of g(x)and h(x):
g′(x)=6x2+ 10x−4and h′(x)=2x+ 3
Applying the quotient rule:
f′(x) = (6x2+ 10x−4)(x2+ 3x)−(2x3+ 5x2−4x+ 7)(2x+ 3)
(x2+ 3x)2
Simplifying the expression gives the derivative of the function f(x).
Feel free to reach out if you need more assistance!
Question 17
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
14
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need ad-
ditional explanations, feel free to ask.Sure! Here is a question on
Calculus for Liberty University along with the step-by-step solutions
in LateX code:
Question 17: Find the derivative of the function f(x)=3x2+ 5x+ 7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x+ 7, we will use the power rule for differentiation.
Step 1: Identify the terms in the function.
Given function: f(x)=3x2+ 5x+ 7
Step 2: Apply the power rule for differentiation to each term.
f′(x) = d
dx (3x2) + d
dx (5x) + d
dx (7)
Step 3: Differentiate each term with respect to x.
f′(x)=2·3x2−1+ 5 ·1x1−1+ 0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 7 is
f′(x) = 6x+ 5.
I hope this helps! If you have any further questions or need addi-
tional explanations, feel free to ask.
Question 18
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
15
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 18:
Find the derivative of the function f(x) = 3x4−8x3+ 2x−5.
Step-by-step Solution:
To find the derivative of the function f(x) = 3x4−8x3+ 2x−5, we
will differentiate each term separately using the power rule.
Given function: f(x)=3x4−8x3+ 2x−5.
1. Differentiate the term 3x4with respect to x:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate the term −8x3with respect to x:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Differentiate the term 2xwith respect to x:
d
dx (2x)=2
4. Since the derivative of a constant is zero, the derivative of the
constant term −5is 0.
Therefore, the derivative of the function f(x)=3x4−8x3+ 2x−5is:
f′(x) = 12x3−24x2+ 2
Question 19
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
16
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.Sure! Here is a calculus question along with its step-by-
step solution in LateX code:
Question 19: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we differentiate each term with respect to xusing the power rule for
derivatives.
df
dx =d
dx (3x2)−d
dx (4x) + d
dx (2)
Applying the power rule, we get:
df
dx = 3 ·2x−4·1+0
Simplifying further:
df
dx = 6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 2 is
f′(x) = 6x−4.
Question 20
Question 20: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x)=3x2−4x+ 2,
we will use the power rule for differentiation which states that the
derivative of xnis nxn−1.
Therefore, the derivative of f(x)=3x2−4x+ 2 is:
d
dx (3x2−4x+ 2) = 3(2)x2−1−4(1)x1−1+ 0
d
dx (3x2−4x+ 2) = 6x−4
17
Hence, the derivative of the function f(x)=3x2−4x+2 is 6x−4.Sure,
here is a Calculus question along with a step-by-step solution in LateX
code :
Question 20: Find the derivative of the function f(x)=3x2−4x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 2,
we will use the power rule for differentiation which states that the
derivative of xnis nxn−1.
Therefore, the derivative of f(x)=3x2−4x+ 2 is:
d
dx (3x2−4x+ 2) = 3(2)x2−1−4(1)x1−1+ 0
d
dx (3x2−4x+ 2) = 6x−4
Hence, the derivative of the function f(x)=3x2−4x+ 2 is 6x−4.
18
