MATH 100 - FUNDAMENTALS OF
MATHEMATICS - Calculus Question Bank - Set
2
Question 1
Question 1: 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
differentiate each term with respect to xusing the power rule for derivatives.
Step 1: Find the derivative of the first term 3x2using the power rule:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of the second term 2xusing the power rule:
d
dx (2x)=1·2x1−1= 2
Step 3: Find the derivative of the constant term −5: Since the derivative of
a constant is 0, the derivative of the constant term −5 is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−5 is:
f′(x)=6x+ 2
This is the solution to question 1.Sure, here is a Calculus question
along with its solution in LateX code:
Question 1: 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 differentiate each term with respect to xusing the power rule
for derivatives.
Step 1: Find the derivative of the first term 3x2using the power
rule: d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of the second term 2xusing the power
rule: d
dx (2x)=1·2x1−1= 2
1
Step 3: Find the derivative of the constant term −5: Since the
derivative of a constant is 0, the derivative of the constant term −5
is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the solution to question 1.
Question 2
Find the derivative of the function f(x) = 3x4−6x3+ 2x2−5x+ 1.
Solution:
To find the derivative of the function f(x)=3x4−6x3+ 2x2−5x+ 1,
we will apply the power rule for differentiation to each term.
Given function: f(x)=3x4−6x3+ 2x2−5x+ 1
The derivative of the function is calculated term by term as follows:
1. For the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. For the term −6x3:
d
dx (−6x3) = 3 ·(−6)x3−1=−18x2
3. For the term 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. For the term −5x:
d
dx (−5x)=5·(−5)x1−1=−5
5. For the term 1: Since the derivative of a constant is zero, we
have: d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−6x3+2x2−5x+1
is:
f′(x) = 12x3−18x2+ 4x−5
Question 2:
Find the derivative of the function f(x) = 3x4−6x3+ 2x2−5x+ 1.
Solution:
To find the derivative of the function f(x)=3x4−6x3+ 2x2−5x+ 1,
we will apply the power rule for differentiation to each term.
2
Given function: f(x)=3x4−6x3+ 2x2−5x+ 1
The derivative of the function is calculated term by term as follows:
1. For the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. For the term −6x3:
d
dx (−6x3) = 3 ·(−6)x3−1=−18x2
3. For the term 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. For the term −5x:
d
dx (−5x)=5·(−5)x1−1=−5
5. For the term 1: Since the derivative of a constant is zero, we
have: d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−6x3+2x2−5x+1
is:
f′(x) = 12x3−18x2+ 4x−5
Question 3
Step-by-step solution: Step 1: Apply the power rule to each term
of the function:
f′(x)=4·3x4−1−3·2x3−1+ 0 + 0
f′(x) = 12x3−6x2
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−7
is f′(x) = 12x3−6x2.Question 3: Find the derivative of the function
f(x)=3x4−2x3+ 5x−7.
Step-by-step solution: Step 1: Apply the power rule to each term
of the function:
f′(x)=4·3x4−1−3·2x3−1+ 0 + 0
f′(x) = 12x3−6x2
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−7is
f′(x) = 12x3−6x2.
3
Question 4
Question 4: Calculate the derivative of the function f(x)=3x4−
2x3+ 5x2−7x+ 1.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 1, 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 (1)
= 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+1
is f′(x) = 12x3−6x2+ 10x−7.Certainly! Here is a question on Calculus
along with the step-by-step solution in LaTeX code:
Question 4: Calculate the derivative of the function f(x) = 3x4−
2x3+ 5x2−7x+ 1.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 1, 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 (1)
= 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+1
is f′(x) = 12x3−6x2+ 10x−7.
Question 5
Find the derivative of the function f(x) = 3x2+ 2√x−1
x.
Step-by-step solution:
Given function: f(x)=3x2+ 2√x−1
x.
To find the derivative of the function f(x), we will differentiate
each term with respect to xusing the properties of differentiation.
1. Find the derivative of the term 3x2:
d
dx (3x2)=2×3x2−1= 6x
.
2. Find the derivative of the term 2√x: We can rewrite 2√xas
2x1/2.d
dx (2x1/2) = 1
2×2x(1/2)−1=1
2×2√x=√x
4
.
3. Find the derivative of the term 1
x:
d
dx 1
x=−1
x2
.
Therefore, the derivative of the function f(x)=3x2+ 2√x−1
xis:
f′(x)=6x+√x+−1
x2= 6x+√x−1
x2
.Question 5:
Find the derivative of the function f(x) = 3x2+ 2√x−1
x.
Step-by-step solution:
Given function: f(x)=3x2+ 2√x−1
x.
To find the derivative of the function f(x), we will differentiate
each term with respect to xusing the properties of differentiation.
1. Find the derivative of the term 3x2:
d
dx (3x2)=2×3x2−1= 6x
.
2. Find the derivative of the term 2√x: We can rewrite 2√xas
2x1/2.d
dx (2x1/2) = 1
2×2x(1/2)−1=1
2×2√x=√x
.
3. Find the derivative of the term 1
x:
d
dx 1
x=−1
x2
.
Therefore, the derivative of the function f(x)=3x2+ 2√x−1
xis:
f′(x)=6x+√x+−1
x2= 6x+√x−1
x2
.
Question 6
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+5x2−7x+9, we apply the power rule for differentiation where
the derivative of xnwith respect to x is nx(n−1).
5
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term like 9 is 0.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 6:
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+5x2−7x+9, we apply the power rule for differentiation where
the derivative of xnwith respect to x is nx(n−1).
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term like 9 is 0.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
6
Question 7
Step-by-step solution: Given function: f(x)=3x2−4x+ 2
To find the derivative of f(x), we will differentiate each term of
the function separately using the power rule.
1. Differentiate 3x2:
d
dx (3x2) = 2(3)x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: Since the derivative of a
constant is 0, the derivative of the term 2is 0.
Therefore, the derivative of the function f(x)is:
f′(x)=6x−4
The derivative of the function f(x)=3x2−4x+ 2 is f′(x)=6x−
4.Question 7: Find the derivative of the function f(x)=3x2−4x+ 2.
Step-by-step solution: Given function: f(x)=3x2−4x+ 2
To find the derivative of f(x), we will differentiate each term of
the function separately using the power rule.
1. Differentiate 3x2:
d
dx (3x2) = 2(3)x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: Since the derivative of a
constant is 0, the derivative of the term 2is 0.
Therefore, the derivative of the function f(x)is:
f′(x)=6x−4
The derivative of the function f(x)=3x2−4x+ 2 is f′(x) = 6x−4.
Question 8
Find the derivative of the following function using the chain rule:
f(x) = p2x3+ 4x+ 1
7
Solution:
To find the derivative of f(x), we first need to rewrite the function
using the chain rule.
Let u= 2x3+ 4x+ 1.
Then, f(x) = √u.
Now, we can find the derivative as follows:
f′(x) = d
dx √u
=d
du √u·du
dx
=1
2√u·(6x2+ 4)
=6x2+ 4
2√2x3+ 4x+ 1
=3x2+ 2
√2x3+ 4x+ 1
Therefore, the derivative of f(x)is 3x2+2
√2x3+4x+1 .Question 8:
Find the derivative of the following function using the chain rule:
f(x) = p2x3+ 4x+ 1
Solution:
To find the derivative of f(x), we first need to rewrite the function
using the chain rule.
Let u= 2x3+ 4x+ 1.
Then, f(x) = √u.
Now, we can find the derivative as follows:
f′(x) = d
dx √u
=d
du √u·du
dx
=1
2√u·(6x2+ 4)
=6x2+ 4
2√2x3+ 4x+ 1
=3x2+ 2
√2x3+ 4x+ 1
Therefore, the derivative of f(x)is 3x2+2
√2x3+4x+1 .
8
Question 9
Question 9: Find the derivative of the following function:
f(x) = 5x3−2x2+ 7x−4
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−4, we need to differentiate each term individually using the power
rule of differentiation.
Given function: f(x)=5x3−2x2+ 7x−4
Step 1: Differentiate 5x3with respect to x:
d
dx (5x3)=3×5x3−1= 15x2
Step 2: Differentiate −2x2with respect to x:
d
dx (−2x2)=2×(−2)x2−1=−4x
Step 3: Differentiate 7xwith respect to x:
d
dx (7x)=7
Step 4: The derivative of a constant term −4is 0.
Therefore, the derivative of the function f(x)is:
f′(x) = 15x2−4x+ 7
Thus, the derivative of the function f(x) = 5x3−2x2+ 7x−4is
f′(x) = 15x2−4x+ 7.Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 9: Find the derivative of the following function:
f(x) = 5x3−2x2+ 7x−4
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−4, we need to differentiate each term individually using the power
rule of differentiation.
Given function: f(x)=5x3−2x2+ 7x−4
Step 1: Differentiate 5x3with respect to x:
d
dx (5x3)=3×5x3−1= 15x2
Step 2: Differentiate −2x2with respect to x:
d
dx (−2x2)=2×(−2)x2−1=−4x
Step 3: Differentiate 7xwith respect to x:
d
dx (7x)=7
9
Step 4: The derivative of a constant term −4is 0.
Therefore, the derivative of the function f(x)is:
f′(x) = 15x2−4x+ 7
Thus, the derivative of the function f(x) = 5x3−2x2+ 7x−4is
f′(x) = 15x2−4x+ 7.
Question 10
Step-by-step solution: Given function: f(x)=3x2−2x+ 5
To find the derivative of f(x)with respect to xusing the power
rule, we differentiate each term of the function separately.
1. Differentiate 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate −2x:
d
dx (−2x) = −2·1x1−1=−2
3. Differentiate the constant term 5: Since the derivative of a
constant is 0,
d
dx (5) = 0
Therefore, the derivative of f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 10: Find the derivative of the function f(x)=3x2−2x+ 5
using the power rule.
Step-by-step solution: Given function: f(x)=3x2−2x+ 5
To find the derivative of f(x)with respect to xusing the power
rule, we differentiate each term of the function separately.
1. Differentiate 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate −2x:
d
dx (−2x) = −2·1x1−1=−2
3. Differentiate the constant term 5: Since the derivative of a
constant is 0,
d
dx (5) = 0
Therefore, the derivative of f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
10
Question 11
Step-by-step solution: To calculate the derivative of the function
f(x) = 3x2+ 2x−4, we will use the power rule for derivatives which
states that the derivative of xnis nxn−1.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of 2x:
d
dx (2x)=2
3. Find the derivative of the constant term −4: Since the derivative
of a constant is zero, the derivative of −4is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−4is:
d
dx (3x2+ 2x−4) = 6x+ 2
Question 11: Calculate the derivative of the following function: f(x) =
3x2+ 2x−4.
Step-by-step solution: To calculate the derivative of the function
f(x) = 3x2+ 2x−4, we will use the power rule for derivatives which
states that the derivative of xnis nxn−1.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of 2x:
d
dx (2x)=2
3. Find the derivative of the constant term −4: Since the derivative
of a constant is zero, the derivative of −4is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−4is:
d
dx (3x2+ 2x−4) = 6x+ 2
Question 12
Step-by-step solution: To find the derivative of the given function,
we need to apply the power rule, which states that the derivative of
xnis n·xn−1.
Given function: f(x)=3x4−2x3+ 5x2−7
1. Find the derivative of 3x4:d
dx (3x4)=4·3x4−1= 12x3
11
2. Find the derivative of −2x3:d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of the constant term −7: 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+ 10xQuestion 12: Calculate the derivative of the
function f(x) = 3x4−2x3+ 5x2−7.
Step-by-step solution: To find the derivative of the given function,
we need to apply the power rule, which states that the derivative of
xnis n·xn−1.
Given function: f(x)=3x4−2x3+ 5x2−7
1. Find the derivative of 3x4:d
dx (3x4)=4·3x4−1= 12x3
2. Find the derivative of −2x3:d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of the constant term −7: 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
Question 13
Question 13: Find the derivative of the following function: f(x) =
4x3−2x2+ 5x−1
Solution: To find the derivative of the function f(x) = 4x3−2x2+
5x−1, we will differentiate each term with respect to xusing the
power rule for derivatives.
Step 1: Find the derivative of the first term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Step 2: Find the derivative of the second term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
Step 3: Find the derivative of the third term 5x:
d
dx (5x)=5
Step 4: Find the derivative of the fourth term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−1is:
f′(x) = 12x2−4x+ 5
12
This is the final answer.
Feel free to reach out if you need any more assistance or have
any other questions!Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 13: Find the derivative of the following function: f(x) =
4x3−2x2+ 5x−1
Solution: To find the derivative of the function f(x) = 4x3−2x2+
5x−1, we will differentiate each term with respect to xusing the
power rule for derivatives.
Step 1: Find the derivative of the first term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Step 2: Find the derivative of the second term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
Step 3: Find the derivative of the third term 5x:
d
dx (5x)=5
Step 4: Find the derivative of the fourth term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−1is:
f′(x) = 12x2−4x+ 5
This is the final answer.
Feel free to reach out if you need any more assistance or have any
other questions!
Question 14
Question 14: Find the derivative of the following function using
the product rule:
f(x) = (3x2+ 5)(e2x)
Step-by-step solution: To find the derivative of the given function
f(x) = (3x2+ 5)(e2x)using the product rule, we need to differentiate
each term and then apply the product rule.
Let’s denote the first function as u= 3x2+5 and the second function
as v=e2x.
13
1. Calculate du
dx , the derivative of u:
du
dx =d
dx (3x2+ 5) = 6x
2. Calculate dv
dx , the derivative of v:
dv
dx =d
dx (e2x) = 2e2x
3. Apply the product rule, which states that the derivative of the
product of two functions is the first function times the derivative of
the second function plus the second function times the derivative of
the first function: d
dx (uv) = u·dv
dx +v·du
dx
4. Substitute the values of u, du
dx , v, and dv
dx into the product rule
formula:
d
dx ((3x2+ 5)(e2x)) = (3x2+ 5) ·(2e2x)+(e2x)·(6x)
5. Simplify the expression:
d
dx ((3x2+ 5)(e2x)) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Therefore, the derivative of the function f(x) = (3x2+ 5)(e2x)using
the product rule is:
f′(x) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Sure! Here is a question on Calculus for Liberty University along
with its step-by-step solution in LateX code:
Question 14: Find the derivative of the following function using
the product rule:
f(x) = (3x2+ 5)(e2x)
Step-by-step solution: To find the derivative of the given function
f(x) = (3x2+ 5)(e2x)using the product rule, we need to differentiate
each term and then apply the product rule.
Let’s denote the first function as u= 3x2+5 and the second function
as v=e2x.
1. Calculate du
dx , the derivative of u:
du
dx =d
dx (3x2+ 5) = 6x
2. Calculate dv
dx , the derivative of v:
dv
dx =d
dx (e2x) = 2e2x
14
3. Apply the product rule, which states that the derivative of the
product of two functions is the first function times the derivative of
the second function plus the second function times the derivative of
the first function: d
dx (uv) = u·dv
dx +v·du
dx
4. Substitute the values of u, du
dx , v, and dv
dx into the product rule
formula:
d
dx ((3x2+ 5)(e2x)) = (3x2+ 5) ·(2e2x)+(e2x)·(6x)
5. Simplify the expression:
d
dx ((3x2+ 5)(e2x)) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Therefore, the derivative of the function f(x) = (3x2+ 5)(e2x)using
the product rule is:
f′(x) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Question 15
Step-by-step solution: To find the derivative of the function f(x),
we need to apply the power rule for differentiation. The power rule
states that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking derivative term by term: f′(x) = d
dx (3x4)−d
dx (2x3)+ d
dx (5x2)−
d
dx (7x) + d
dx (2)
Applying the power rule: f′(x) = 3(4)x4−1−2(3)x3−1+ 5(2)x2−1−
7(1)x1−1+ 0
Simplifying: 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 15: Find the derivative of the
function f(x) = 3x4−2x3+ 5x2−7x+ 2.
Step-by-step solution: To find the derivative of the function f(x),
we need to apply the power rule for differentiation. The power rule
states that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking derivative term by term: f′(x) = d
dx (3x4)−d
dx (2x3)+ d
dx (5x2)−
d
dx (7x) + d
dx (2)
Applying the power rule: f′(x) = 3(4)x4−1−2(3)x3−1+ 5(2)x2−1−
7(1)x1−1+ 0
Simplifying: 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.
15
Question 16
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9.
Solution:
To find the derivative of f(x) = 3x4−2x3+ 5x2−7x+ 9, we will use
the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
1. Find the derivative of each term:
d
dx (3x4)=4·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7x)=1·(−7)x1−1=−7
d
dx (9) = 0
2. Combine the derivatives of each term to find the derivative of
the function:
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 16:
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9.
Solution:
To find the derivative of f(x) = 3x4−2x3+ 5x2−7x+ 9, we will use
the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
1. Find the derivative of each term:
d
dx (3x4)=4·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7x)=1·(−7)x1−1=−7
d
dx (9) = 0
2. Combine the derivatives of each term to find the derivative of
the function:
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 17
Question 17: 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 use the power rule of differentiation. The power
16
rule states that for any constant n, the derivative of xnwith respect
to xis nxn−1.
Given function: f(x)=3x4+ 2x2+ 5
Taking the derivative term by term:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (2x2)=2·2x2−1= 4x
d
dx (5) = 0
(derivative of a constant is 0)
Therefore, the derivative of the function f(x)=3x4+ 2x2+ 5 is:
df(x)
dx = 12x3+ 4x
This completes the solution.Sure! Here is a Calculus question
along with the step-by-step solution in LateX code:
Question 17: 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 use the power rule of differentiation. The power
rule states that for any constant n, the derivative of xnwith respect
to xis nxn−1.
Given function: f(x)=3x4+ 2x2+ 5
Taking the derivative term by term:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (2x2)=2·2x2−1= 4x
d
dx (5) = 0
(derivative of a constant is 0)
Therefore, the derivative of the function f(x)=3x4+ 2x2+ 5 is:
df(x)
dx = 12x3+ 4x
This completes the solution.
17
Question 18
Question 18: Compute the following limit:
lim
x→3
x2−9
x−3
Solution:
To find the limit of the given function as xapproaches 3, we can
try simplifying the function first by factoring the numerator.
lim
x→3
x2−9
x−3= lim
x→3
(x−3)(x+ 3)
x−3
Now, we can cancel out the common factor of x−3from the nu-
merator and denominator:
= lim
x→3(x+ 3)
Finally, substitute x= 3 into the expression to find the limit:
= 3 + 3 = 6
Therefore, the limit of the given function as xapproaches 3 is 6.
Feel free to reach out if you need any more help or have any
other questions!Certainly! Here is a Calculus question along with its
solution in LateX code:
Question 18: Compute the following limit:
lim
x→3
x2−9
x−3
Solution:
To find the limit of the given function as xapproaches 3, we can
try simplifying the function first by factoring the numerator.
lim
x→3
x2−9
x−3= lim
x→3
(x−3)(x+ 3)
x−3
Now, we can cancel out the common factor of x−3from the nu-
merator and denominator:
= lim
x→3(x+ 3)
Finally, substitute x= 3 into the expression to find the limit:
= 3 + 3 = 6
Therefore, the limit of the given function as xapproaches 3 is 6.
Feel free to reach out if you need any more help or have any other
questions!
18
Question 19
Question 19: Compute the definite integral R2
0(3x2−4x+ 5) dx.
Solution: To compute the definite integral, we will integrate term
by term and then evaluate the integral at the upper and lower limits.
Z2
0
(3x2−4x+ 5) dx =Z2
0
3x2dx −Z2
0
4x dx +Z2
0
5dx
Now, integrating each term:
Z2
0
3x2dx =x32
0= 23−03= 8
Z2
0
4x dx =2x22
0= 2(2)2−2(0)2= 8
Z2
0
5dx = [5x]2
0= 5(2) −5(0) = 10
Therefore, the definite integral R2
0(3x2−4x+ 5) dx is:
Z2
0
(3x2−4x+ 5) dx = 8 −8 + 10 = 10
Sure! Here’s a Calculus question along with its step-by-step solution
in LateX code:
Question 19: Compute the definite integral R2
0(3x2−4x+ 5) dx.
Solution: To compute the definite integral, we will integrate term
by term and then evaluate the integral at the upper and lower limits.
Z2
0
(3x2−4x+ 5) dx =Z2
0
3x2dx −Z2
0
4x dx +Z2
0
5dx
Now, integrating each term:
Z2
0
3x2dx =x32
0= 23−03= 8
Z2
0
4x dx =2x22
0= 2(2)2−2(0)2= 8
Z2
0
5dx = [5x]2
0= 5(2) −5(0) = 10
Therefore, the definite integral R2
0(3x2−4x+ 5) dx is:
Z2
0
(3x2−4x+ 5) dx = 8 −8 + 10 = 10
19
Question 20
Z(4x3−6x2+ 2x−9) dx
Step-by-step solution: We will use the power rule for integration
to calculate the indefinite integral of each term separately:
Z(4x3−6x2+ 2x−9) dx =Z4x3dx −Z6x2dx +Z2x dx −Z9dx
Integrating each term:
=4
4x4−6
3x3+2
2x−9x+C
Simplifying:
=x4−2x3+x−9x+C
Therefore, the indefinite integral R(4x3−6x2+ 2x−9) dx is equal to
x4−2x3+x−9x+C, where Cis the constant of integration.Question
20: Calculate the following indefinite integral:
Z(4x3−6x2+ 2x−9) dx
Step-by-step solution: We will use the power rule for integration
to calculate the indefinite integral of each term separately:
Z(4x3−6x2+ 2x−9) dx =Z4x3dx −Z6x2dx +Z2x dx −Z9dx
Integrating each term:
=4
4x4−6
3x3+2
2x−9x+C
Simplifying:
=x4−2x3+x−9x+C
Therefore, the indefinite integral R(4x3−6x2+ 2x−9) dx is equal to
x4−2x3+x−9x+C, where Cis the constant of integration.
20
Step 3: Find the derivative of the constant term −5: Since the
derivative of a constant is 0, the derivative of the constant term −5
is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−5is:
f′(x)=6x+ 2
This is the solution to question 1.
Question 2
Find the derivative of the function f(x) = 3x4−6x3+ 2x2−5x+ 1.
Solution:
To find the derivative of the function f(x)=3x4−6x3+ 2x2−5x+ 1,
we will apply the power rule for differentiation to each term.
Given function: f(x)=3x4−6x3+ 2x2−5x+ 1
The derivative of the function is calculated term by term as follows:
1. For the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. For the term −6x3:
d
dx (−6x3) = 3 ·(−6)x3−1=−18x2
3. For the term 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. For the term −5x:
d
dx (−5x)=5·(−5)x1−1=−5
5. For the term 1: Since the derivative of a constant is zero, we
have: d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−6x3+2x2−5x+1
is:
f′(x) = 12x3−18x2+ 4x−5
Question 2:
Find the derivative of the function f(x) = 3x4−6x3+ 2x2−5x+ 1.
Solution:
To find the derivative of the function f(x)=3x4−6x3+ 2x2−5x+ 1,
we will apply the power rule for differentiation to each term.
2
Given function: f(x)=3x4−6x3+ 2x2−5x+ 1
The derivative of the function is calculated term by term as follows:
1. For the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. For the term −6x3:
d
dx (−6x3) = 3 ·(−6)x3−1=−18x2
3. For the term 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. For the term −5x:
d
dx (−5x)=5·(−5)x1−1=−5
5. For the term 1: Since the derivative of a constant is zero, we
have: d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−6x3+2x2−5x+1
is:
f′(x) = 12x3−18x2+ 4x−5
Question 3
Step-by-step solution: Step 1: Apply the power rule to each term
of the function:
f′(x)=4·3x4−1−3·2x3−1+ 0 + 0
f′(x) = 12x3−6x2
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−7
is f′(x) = 12x3−6x2.Question 3: Find the derivative of the function
f(x)=3x4−2x3+ 5x−7.
Step-by-step solution: Step 1: Apply the power rule to each term
of the function:
f′(x)=4·3x4−1−3·2x3−1+ 0 + 0
f′(x) = 12x3−6x2
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−7is
f′(x) = 12x3−6x2.
3
Question 4
Question 4: Calculate the derivative of the function f(x)=3x4−
2x3+ 5x2−7x+ 1.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 1, 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 (1)
= 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+1
is f′(x) = 12x3−6x2+ 10x−7.Certainly! Here is a question on Calculus
along with the step-by-step solution in LaTeX code:
Question 4: Calculate the derivative of the function f(x) = 3x4−
2x3+ 5x2−7x+ 1.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 1, 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 (1)
= 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+1
is f′(x) = 12x3−6x2+ 10x−7.
Question 5
Find the derivative of the function f(x) = 3x2+ 2√x−1
x.
Step-by-step solution:
Given function: f(x)=3x2+ 2√x−1
x.
To find the derivative of the function f(x), we will differentiate
each term with respect to xusing the properties of differentiation.
1. Find the derivative of the term 3x2:
d
dx (3x2)=2×3x2−1= 6x
.
2. Find the derivative of the term 2√x: We can rewrite 2√xas
2x1/2.d
dx (2x1/2) = 1
2×2x(1/2)−1=1
2×2√x=√x
4
.
3. Find the derivative of the term 1
x:
d
dx 1
x=−1
x2
.
Therefore, the derivative of the function f(x)=3x2+ 2√x−1
xis:
f′(x)=6x+√x+−1
x2= 6x+√x−1
x2
.Question 5:
Find the derivative of the function f(x) = 3x2+ 2√x−1
x.
Step-by-step solution:
Given function: f(x)=3x2+ 2√x−1
x.
To find the derivative of the function f(x), we will differentiate
each term with respect to xusing the properties of differentiation.
1. Find the derivative of the term 3x2:
d
dx (3x2)=2×3x2−1= 6x
.
2. Find the derivative of the term 2√x: We can rewrite 2√xas
2x1/2.d
dx (2x1/2) = 1
2×2x(1/2)−1=1
2×2√x=√x
.
3. Find the derivative of the term 1
x:
d
dx 1
x=−1
x2
.
Therefore, the derivative of the function f(x)=3x2+ 2√x−1
xis:
f′(x)=6x+√x+−1
x2= 6x+√x−1
x2
.
Question 6
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+5x2−7x+9, we apply the power rule for differentiation where
the derivative of xnwith respect to x is nx(n−1).
5
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term like 9 is 0.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 6:
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+5x2−7x+9, we apply the power rule for differentiation where
the derivative of xnwith respect to x is nx(n−1).
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term like 9 is 0.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
6
Question 7
Step-by-step solution: Given function: f(x)=3x2−4x+ 2
To find the derivative of f(x), we will differentiate each term of
the function separately using the power rule.
1. Differentiate 3x2:
d
dx (3x2) = 2(3)x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: Since the derivative of a
constant is 0, the derivative of the term 2is 0.
Therefore, the derivative of the function f(x)is:
f′(x)=6x−4
The derivative of the function f(x)=3x2−4x+ 2 is f′(x)=6x−
4.Question 7: Find the derivative of the function f(x)=3x2−4x+ 2.
Step-by-step solution: Given function: f(x)=3x2−4x+ 2
To find the derivative of f(x), we will differentiate each term of
the function separately using the power rule.
1. Differentiate 3x2:
d
dx (3x2) = 2(3)x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 2: Since the derivative of a
constant is 0, the derivative of the term 2is 0.
Therefore, the derivative of the function f(x)is:
f′(x)=6x−4
The derivative of the function f(x)=3x2−4x+ 2 is f′(x) = 6x−4.
Question 8
Find the derivative of the following function using the chain rule:
f(x) = p2x3+ 4x+ 1
7
Solution:
To find the derivative of f(x), we first need to rewrite the function
using the chain rule.
Let u= 2x3+ 4x+ 1.
Then, f(x) = √u.
Now, we can find the derivative as follows:
f′(x) = d
dx √u
=d
du √u·du
dx
=1
2√u·(6x2+ 4)
=6x2+ 4
2√2x3+ 4x+ 1
=3x2+ 2
√2x3+ 4x+ 1
Therefore, the derivative of f(x)is 3x2+2
√2x3+4x+1 .Question 8:
Find the derivative of the following function using the chain rule:
f(x) = p2x3+ 4x+ 1
Solution:
To find the derivative of f(x), we first need to rewrite the function
using the chain rule.
Let u= 2x3+ 4x+ 1.
Then, f(x) = √u.
Now, we can find the derivative as follows:
f′(x) = d
dx √u
=d
du √u·du
dx
=1
2√u·(6x2+ 4)
=6x2+ 4
2√2x3+ 4x+ 1
=3x2+ 2
√2x3+ 4x+ 1
Therefore, the derivative of f(x)is 3x2+2
√2x3+4x+1 .
8
Question 9
Question 9: Find the derivative of the following function:
f(x) = 5x3−2x2+ 7x−4
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−4, we need to differentiate each term individually using the power
rule of differentiation.
Given function: f(x)=5x3−2x2+ 7x−4
Step 1: Differentiate 5x3with respect to x:
d
dx (5x3)=3×5x3−1= 15x2
Step 2: Differentiate −2x2with respect to x:
d
dx (−2x2)=2×(−2)x2−1=−4x
Step 3: Differentiate 7xwith respect to x:
d
dx (7x)=7
Step 4: The derivative of a constant term −4is 0.
Therefore, the derivative of the function f(x)is:
f′(x) = 15x2−4x+ 7
Thus, the derivative of the function f(x) = 5x3−2x2+ 7x−4is
f′(x) = 15x2−4x+ 7.Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 9: Find the derivative of the following function:
f(x) = 5x3−2x2+ 7x−4
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−4, we need to differentiate each term individually using the power
rule of differentiation.
Given function: f(x)=5x3−2x2+ 7x−4
Step 1: Differentiate 5x3with respect to x:
d
dx (5x3)=3×5x3−1= 15x2
Step 2: Differentiate −2x2with respect to x:
d
dx (−2x2)=2×(−2)x2−1=−4x
Step 3: Differentiate 7xwith respect to x:
d
dx (7x)=7
9
Step 4: The derivative of a constant term −4is 0.
Therefore, the derivative of the function f(x)is:
f′(x) = 15x2−4x+ 7
Thus, the derivative of the function f(x) = 5x3−2x2+ 7x−4is
f′(x) = 15x2−4x+ 7.
Question 10
Step-by-step solution: Given function: f(x)=3x2−2x+ 5
To find the derivative of f(x)with respect to xusing the power
rule, we differentiate each term of the function separately.
1. Differentiate 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate −2x:
d
dx (−2x) = −2·1x1−1=−2
3. Differentiate the constant term 5: Since the derivative of a
constant is 0,
d
dx (5) = 0
Therefore, the derivative of f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 10: Find the derivative of the function f(x)=3x2−2x+ 5
using the power rule.
Step-by-step solution: Given function: f(x)=3x2−2x+ 5
To find the derivative of f(x)with respect to xusing the power
rule, we differentiate each term of the function separately.
1. Differentiate 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate −2x:
d
dx (−2x) = −2·1x1−1=−2
3. Differentiate the constant term 5: Since the derivative of a
constant is 0,
d
dx (5) = 0
Therefore, the derivative of f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
10
Question 11
Step-by-step solution: To calculate the derivative of the function
f(x) = 3x2+ 2x−4, we will use the power rule for derivatives which
states that the derivative of xnis nxn−1.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of 2x:
d
dx (2x)=2
3. Find the derivative of the constant term −4: Since the derivative
of a constant is zero, the derivative of −4is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−4is:
d
dx (3x2+ 2x−4) = 6x+ 2
Question 11: Calculate the derivative of the following function: f(x) =
3x2+ 2x−4.
Step-by-step solution: To calculate the derivative of the function
f(x) = 3x2+ 2x−4, we will use the power rule for derivatives which
states that the derivative of xnis nxn−1.
1. Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Find the derivative of 2x:
d
dx (2x)=2
3. Find the derivative of the constant term −4: Since the derivative
of a constant is zero, the derivative of −4is 0.
Therefore, the derivative of the function f(x)=3x2+ 2x−4is:
d
dx (3x2+ 2x−4) = 6x+ 2
Question 12
Step-by-step solution: To find the derivative of the given function,
we need to apply the power rule, which states that the derivative of
xnis n·xn−1.
Given function: f(x)=3x4−2x3+ 5x2−7
1. Find the derivative of 3x4:d
dx (3x4)=4·3x4−1= 12x3
11
2. Find the derivative of −2x3:d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of the constant term −7: 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+ 10xQuestion 12: Calculate the derivative of the
function f(x) = 3x4−2x3+ 5x2−7.
Step-by-step solution: To find the derivative of the given function,
we need to apply the power rule, which states that the derivative of
xnis n·xn−1.
Given function: f(x)=3x4−2x3+ 5x2−7
1. Find the derivative of 3x4:d
dx (3x4)=4·3x4−1= 12x3
2. Find the derivative of −2x3:d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Find the derivative of 5x2:d
dx (5x2)=2·5x2−1= 10x
4. Find the derivative of the constant term −7: 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
Question 13
Question 13: Find the derivative of the following function: f(x) =
4x3−2x2+ 5x−1
Solution: To find the derivative of the function f(x) = 4x3−2x2+
5x−1, we will differentiate each term with respect to xusing the
power rule for derivatives.
Step 1: Find the derivative of the first term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Step 2: Find the derivative of the second term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
Step 3: Find the derivative of the third term 5x:
d
dx (5x)=5
Step 4: Find the derivative of the fourth term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−1is:
f′(x) = 12x2−4x+ 5
12
This is the final answer.
Feel free to reach out if you need any more assistance or have
any other questions!Sure! Here is a Calculus question along with its
step-by-step solution in LateX code:
Question 13: Find the derivative of the following function: f(x) =
4x3−2x2+ 5x−1
Solution: To find the derivative of the function f(x) = 4x3−2x2+
5x−1, we will differentiate each term with respect to xusing the
power rule for derivatives.
Step 1: Find the derivative of the first term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
Step 2: Find the derivative of the second term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
Step 3: Find the derivative of the third term 5x:
d
dx (5x)=5
Step 4: Find the derivative of the fourth term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−1is:
f′(x) = 12x2−4x+ 5
This is the final answer.
Feel free to reach out if you need any more assistance or have any
other questions!
Question 14
Question 14: Find the derivative of the following function using
the product rule:
f(x) = (3x2+ 5)(e2x)
Step-by-step solution: To find the derivative of the given function
f(x) = (3x2+ 5)(e2x)using the product rule, we need to differentiate
each term and then apply the product rule.
Let’s denote the first function as u= 3x2+5 and the second function
as v=e2x.
13
1. Calculate du
dx , the derivative of u:
du
dx =d
dx (3x2+ 5) = 6x
2. Calculate dv
dx , the derivative of v:
dv
dx =d
dx (e2x) = 2e2x
3. Apply the product rule, which states that the derivative of the
product of two functions is the first function times the derivative of
the second function plus the second function times the derivative of
the first function: d
dx (uv) = u·dv
dx +v·du
dx
4. Substitute the values of u, du
dx , v, and dv
dx into the product rule
formula:
d
dx ((3x2+ 5)(e2x)) = (3x2+ 5) ·(2e2x)+(e2x)·(6x)
5. Simplify the expression:
d
dx ((3x2+ 5)(e2x)) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Therefore, the derivative of the function f(x) = (3x2+ 5)(e2x)using
the product rule is:
f′(x) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Sure! Here is a question on Calculus for Liberty University along
with its step-by-step solution in LateX code:
Question 14: Find the derivative of the following function using
the product rule:
f(x) = (3x2+ 5)(e2x)
Step-by-step solution: To find the derivative of the given function
f(x) = (3x2+ 5)(e2x)using the product rule, we need to differentiate
each term and then apply the product rule.
Let’s denote the first function as u= 3x2+5 and the second function
as v=e2x.
1. Calculate du
dx , the derivative of u:
du
dx =d
dx (3x2+ 5) = 6x
2. Calculate dv
dx , the derivative of v:
dv
dx =d
dx (e2x) = 2e2x
14
3. Apply the product rule, which states that the derivative of the
product of two functions is the first function times the derivative of
the second function plus the second function times the derivative of
the first function: d
dx (uv) = u·dv
dx +v·du
dx
4. Substitute the values of u, du
dx , v, and dv
dx into the product rule
formula:
d
dx ((3x2+ 5)(e2x)) = (3x2+ 5) ·(2e2x)+(e2x)·(6x)
5. Simplify the expression:
d
dx ((3x2+ 5)(e2x)) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Therefore, the derivative of the function f(x) = (3x2+ 5)(e2x)using
the product rule is:
f′(x) = 6x(3x2+ 5)e2x+ 2(e2x)(3x2+ 5)
Question 15
Step-by-step solution: To find the derivative of the function f(x),
we need to apply the power rule for differentiation. The power rule
states that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking derivative term by term: f′(x) = d
dx (3x4)−d
dx (2x3)+ d
dx (5x2)−
d
dx (7x) + d
dx (2)
Applying the power rule: f′(x) = 3(4)x4−1−2(3)x3−1+ 5(2)x2−1−
7(1)x1−1+ 0
Simplifying: 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 15: Find the derivative of the
function f(x) = 3x4−2x3+ 5x2−7x+ 2.
Step-by-step solution: To find the derivative of the function f(x),
we need to apply the power rule for differentiation. The power rule
states that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 2
Taking derivative term by term: f′(x) = d
dx (3x4)−d
dx (2x3)+ d
dx (5x2)−
d
dx (7x) + d
dx (2)
Applying the power rule: f′(x) = 3(4)x4−1−2(3)x3−1+ 5(2)x2−1−
7(1)x1−1+ 0
Simplifying: 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.
15
Question 16
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9.
Solution:
To find the derivative of f(x) = 3x4−2x3+ 5x2−7x+ 9, we will use
the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
1. Find the derivative of each term:
d
dx (3x4)=4·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7x)=1·(−7)x1−1=−7
d
dx (9) = 0
2. Combine the derivatives of each term to find the derivative of
the function:
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 16:
Find the derivative of the following function: f(x) = 3x4−2x3+
5x2−7x+ 9.
Solution:
To find the derivative of f(x) = 3x4−2x3+ 5x2−7x+ 9, we will use
the power rule for differentiation.
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
1. Find the derivative of each term:
d
dx (3x4)=4·3x4−1= 12x3
d
dx (−2x3)=3·(−2)x3−1=−6x2
d
dx (5x2)=2·5x2−1= 10x
d
dx (−7x)=1·(−7)x1−1=−7
d
dx (9) = 0
2. Combine the derivatives of each term to find the derivative of
the function:
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 17
Question 17: 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 use the power rule of differentiation. The power
16
rule states that for any constant n, the derivative of xnwith respect
to xis nxn−1.
Given function: f(x)=3x4+ 2x2+ 5
Taking the derivative term by term:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (2x2)=2·2x2−1= 4x
d
dx (5) = 0
(derivative of a constant is 0)
Therefore, the derivative of the function f(x)=3x4+ 2x2+ 5 is:
df(x)
dx = 12x3+ 4x
This completes the solution.Sure! Here is a Calculus question
along with the step-by-step solution in LateX code:
Question 17: 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 use the power rule of differentiation. The power
rule states that for any constant n, the derivative of xnwith respect
to xis nxn−1.
Given function: f(x)=3x4+ 2x2+ 5
Taking the derivative term by term:
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (2x2)=2·2x2−1= 4x
d
dx (5) = 0
(derivative of a constant is 0)
Therefore, the derivative of the function f(x)=3x4+ 2x2+ 5 is:
df(x)
dx = 12x3+ 4x
This completes the solution.
17
Question 18
Question 18: Compute the following limit:
lim
x→3
x2−9
x−3
Solution:
To find the limit of the given function as xapproaches 3, we can
try simplifying the function first by factoring the numerator.
lim
x→3
x2−9
x−3= lim
x→3
(x−3)(x+ 3)
x−3
Now, we can cancel out the common factor of x−3from the nu-
merator and denominator:
= lim
x→3(x+ 3)
Finally, substitute x= 3 into the expression to find the limit:
= 3 + 3 = 6
Therefore, the limit of the given function as xapproaches 3 is 6.
Feel free to reach out if you need any more help or have any
other questions!Certainly! Here is a Calculus question along with its
solution in LateX code:
Question 18: Compute the following limit:
lim
x→3
x2−9
x−3
Solution:
To find the limit of the given function as xapproaches 3, we can
try simplifying the function first by factoring the numerator.
lim
x→3
x2−9
x−3= lim
x→3
(x−3)(x+ 3)
x−3
Now, we can cancel out the common factor of x−3from the nu-
merator and denominator:
= lim
x→3(x+ 3)
Finally, substitute x= 3 into the expression to find the limit:
= 3 + 3 = 6
Therefore, the limit of the given function as xapproaches 3 is 6.
Feel free to reach out if you need any more help or have any other
questions!
18
Question 19
Question 19: Compute the definite integral R2
0(3x2−4x+ 5) dx.
Solution: To compute the definite integral, we will integrate term
by term and then evaluate the integral at the upper and lower limits.
Z2
0
(3x2−4x+ 5) dx =Z2
0
3x2dx −Z2
0
4x dx +Z2
0
5dx
Now, integrating each term:
Z2
0
3x2dx =x32
0= 23−03= 8
Z2
0
4x dx =2x22
0= 2(2)2−2(0)2= 8
Z2
0
5dx = [5x]2
0= 5(2) −5(0) = 10
Therefore, the definite integral R2
0(3x2−4x+ 5) dx is:
Z2
0
(3x2−4x+ 5) dx = 8 −8 + 10 = 10
Sure! Here’s a Calculus question along with its step-by-step solution
in LateX code:
Question 19: Compute the definite integral R2
0(3x2−4x+ 5) dx.
Solution: To compute the definite integral, we will integrate term
by term and then evaluate the integral at the upper and lower limits.
Z2
0
(3x2−4x+ 5) dx =Z2
0
3x2dx −Z2
0
4x dx +Z2
0
5dx
Now, integrating each term:
Z2
0
3x2dx =x32
0= 23−03= 8
Z2
0
4x dx =2x22
0= 2(2)2−2(0)2= 8
Z2
0
5dx = [5x]2
0= 5(2) −5(0) = 10
Therefore, the definite integral R2
0(3x2−4x+ 5) dx is:
Z2
0
(3x2−4x+ 5) dx = 8 −8 + 10 = 10
19
Question 20
Z(4x3−6x2+ 2x−9) dx
Step-by-step solution: We will use the power rule for integration
to calculate the indefinite integral of each term separately:
Z(4x3−6x2+ 2x−9) dx =Z4x3dx −Z6x2dx +Z2x dx −Z9dx
Integrating each term:
=4
4x4−6
3x3+2
2x−9x+C
Simplifying:
=x4−2x3+x−9x+C
Therefore, the indefinite integral R(4x3−6x2+ 2x−9) dx is equal to
x4−2x3+x−9x+C, where Cis the constant of integration.Question
20: Calculate the following indefinite integral:
Z(4x3−6x2+ 2x−9) dx
Step-by-step solution: We will use the power rule for integration
to calculate the indefinite integral of each term separately:
Z(4x3−6x2+ 2x−9) dx =Z4x3dx −Z6x2dx +Z2x dx −Z9dx
Integrating each term:
=4
4x4−6
3x3+2
2x−9x+C
Simplifying:
=x4−2x3+x−9x+C
Therefore, the indefinite integral R(4x3−6x2+ 2x−9) dx is equal to
x4−2x3+x−9x+C, where Cis the constant of integration.
20
