MATH 100 - FUNDAMENTALS OF
MATHEMATICS - Calculus Question Bank - Set
3
Question 1
Find the derivative of the function f(x)=3x2+ 5x−2.
Solution:
Given function: f(x)=3x2+ 5x−2
To find the derivative of the function, we use the power rule.
The power rule of differentiation states that if f(x) = axn, then f′(x) =
anxn−1.
Applying the power rule to each term of the function f(x), we get:
f′(x) = 2 ·3x2−1+ 1 ·5x1−1+ 0 = 6x+ 5
Therefore, the derivative of the function f(x)=3x2+ 5x−2 is f′(x) =
6x+ 5.Question 1:
Find the derivative of the function f(x) = 3x2+ 5x−2.
Solution:
Given function: f(x)=3x2+ 5x−2
To find the derivative of the function, we use the power rule.
The power rule of differentiation states that if f(x) = axn, then
f′(x) = anxn−1.
Applying the power rule to each term of the function f(x), we get:
f′(x)=2·3x2−1+ 1 ·5x1−1+ 0 = 6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x−2is
f′(x)=6x+ 5.
Question 2
Compute the derivative of the following function:
f(x)=3x4−5x2+ 2x−7
Solution:
To find the derivative of the function f(x), we will use the power
rule which states that the derivative of xnis nxn−1.
1
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.Question 2:
Compute the derivative of the following function:
f(x)=3x4−5x2+ 2x−7
Solution:
To find the derivative of the function f(x), we will use the power
rule which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.
Question 3
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
2
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−
7.Sure! Here is a Calculus question along with its step-by-step solu-
tion in LateX code:
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−7.
Question 4
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
3
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5Question 4: Find the derivative of the following
function f(x) = 4x3−2x2+ 5x−1.
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5
Question 5
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.Question 5: Find the absolute maximum and absolute minimum
values of the function f(x) = x3−3x2+ 2 on the interval [0,2].
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
4
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.
Question 6
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x) = 0 to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.Question 6: Find the absolute maximum
and absolute minimum values of the function f(x) = x3−3x2+ 2 on
the interval [−1,3].
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x)=0to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
5
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.
Question 7
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 7:
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
6
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 8
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.Certainly! Here’s a Calculus question along with
its step-by-step solution in LateX code:
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.
Question 9
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.Certainly! Here’s a Calculus question along
with the step-by-step solution in LateX code:
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
8
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.
Question 10
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its step-by-step solution in LateX code:
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 11
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a question on Calculus for
Liberty University along with the step-by-step solutions in LaTeX
code:
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
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 12
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
10
Therefore, the derivative of the function f(x) = e3xis 3e3x.Question
12: Find the derivative of the function f(x) = e3xusing the power rule
for derivatives.
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
Therefore, the derivative of the function f(x) = e3xis 3e3x.
Question 13
Question 13:
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
apply the power rule and the constant multiple rule.
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.Sure, here is a calculus question along with its step-by-
step solution in LateX code:
Question 13:
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
apply the power rule and the constant multiple rule.
11
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.
Question 14
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
12
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.Certainly! Here is a calculus question
along with its step-by-step solution in LateX code:
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.
13
Question 15
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.Question 15: Find the derivative of the
function f(x) = 3x4+ 2x3−5x2+ 4x−1using the power rule.
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.
Question 16
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
14
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 16: Find the derivative of 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 differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 17
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
15
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7. Question 17:
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7.
Question 18
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
16
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.Certainly! Here’s a Calculus
question along with its step-by-step solution in LaTeX code:
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.
17
Question 19
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.Question
19:
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.
Question 20
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
to express the integrand as a sum of partial fractions: x2−3x+2
x(x−1) =
A
x+B
x−1. 5. Find the values of Aand Bby equating coefficients
and solving the resulting system of linear equations. 6. The partial
fraction decomposition is x2−3x+2
x(x−1) =1
x−2
x−1. 7. Integrate each partial
fraction separately to get R1
xdx −R2
x−1dx. 8. The antiderivative of
1
xis ln |x|. 9. The antiderivative of 2
x−1is 2 ln |x−1|. 10. Thus,
the indefinite integral of the original function is ln |x| − 2 ln |x−1|+C,
where Cis the constant of integration.Question 20: Find the indefinite
integral Rx2−3x+2
x2−xdx.
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
18
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.Question 2:
Compute the derivative of the following function:
f(x)=3x4−5x2+ 2x−7
Solution:
To find the derivative of the function f(x), we will use the power
rule which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.
Question 3
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
2
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−
7.Sure! Here is a Calculus question along with its step-by-step solu-
tion in LateX code:
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−7.
Question 4
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
3
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5Question 4: Find the derivative of the following
function f(x) = 4x3−2x2+ 5x−1.
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5
Question 5
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.Question 5: Find the absolute maximum and absolute minimum
values of the function f(x) = x3−3x2+ 2 on the interval [0,2].
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
4
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.
Question 6
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x) = 0 to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.Question 6: Find the absolute maximum
and absolute minimum values of the function f(x) = x3−3x2+ 2 on
the interval [−1,3].
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x)=0to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
5
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.
Question 7
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 7:
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
6
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 8
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.Certainly! Here’s a Calculus question along with
its step-by-step solution in LateX code:
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.
Question 9
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.Certainly! Here’s a Calculus question along
with the step-by-step solution in LateX code:
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
8
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.
Question 10
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its step-by-step solution in LateX code:
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 11
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a question on Calculus for
Liberty University along with the step-by-step solutions in LaTeX
code:
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
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 12
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
10
Therefore, the derivative of the function f(x) = e3xis 3e3x.Question
12: Find the derivative of the function f(x) = e3xusing the power rule
for derivatives.
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
Therefore, the derivative of the function f(x) = e3xis 3e3x.
Question 13
Question 13:
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
apply the power rule and the constant multiple rule.
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.Sure, here is a calculus question along with its step-by-
step solution in LateX code:
Question 13:
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
apply the power rule and the constant multiple rule.
11
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.
Question 14
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
12
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.Certainly! Here is a calculus question
along with its step-by-step solution in LateX code:
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.
13
Question 15
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.Question 15: Find the derivative of the
function f(x) = 3x4+ 2x3−5x2+ 4x−1using the power rule.
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.
Question 16
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
14
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 16: Find the derivative of 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 differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 17
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
15
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7. Question 17:
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7.
Question 18
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
16
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.Certainly! Here’s a Calculus
question along with its step-by-step solution in LaTeX code:
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.
17
Question 19
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.Question
19:
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.
Question 20
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
to express the integrand as a sum of partial fractions: x2−3x+2
x(x−1) =
A
x+B
x−1. 5. Find the values of Aand Bby equating coefficients
and solving the resulting system of linear equations. 6. The partial
fraction decomposition is x2−3x+2
x(x−1) =1
x−2
x−1. 7. Integrate each partial
fraction separately to get R1
xdx −R2
x−1dx. 8. The antiderivative of
1
xis ln |x|. 9. The antiderivative of 2
x−1is 2 ln |x−1|. 10. Thus,
the indefinite integral of the original function is ln |x| − 2 ln |x−1|+C,
where Cis the constant of integration.Question 20: Find the indefinite
integral Rx2−3x+2
x2−xdx.
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
18
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.Question 2:
Compute the derivative of the following function:
f(x)=3x4−5x2+ 2x−7
Solution:
To find the derivative of the function f(x), we will use the power
rule which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−5x2+ 2x−7
Step 1: Find the derivative of each term separately using the power
rule.
f′(x) = d
dx (3x4)−d
dx (5x2) + d
dx (2x)−d
dx (7)
= 3 ·4x4−1−5·2x2−1+ 2 ·1x1−1−0
= 12x3−10x+ 2
Therefore, the derivative of the function f(x)=3x4−5x2+ 2x−7is
f′(x) = 12x3−10x+ 2.
Question 3
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
2
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−
7.Sure! Here is a Calculus question along with its step-by-step solu-
tion in LateX code:
Question 3: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 4.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 4, we will differentiate each term with respect to x.
d
dx (3x4) = 12x3
d
dx (−2x3) = −6x2
d
dx (5x2) = 10x
d
dx (−7x) = −7
d
dx (4) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is:
f′(x) = 12x3−6x2+ 10x−7
Hence, the derivative of the function is f′(x) = 12x3−6x2+ 10x−7.
Question 4
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
3
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5Question 4: Find the derivative of the following
function f(x) = 4x3−2x2+ 5x−1.
Step-by-step solution: The derivative of a function can be found
by applying the power rule, which states that the derivative of xnis
nxn−1.
Given function: f(x)=4x3−2x2+ 5x−1
To find the derivative of f(x), we will differentiate each term sep-
arately:
1. Differentiate 4x3:d
dx (4x3)=4·3x3−1= 12x2
2. Differentiate −2x2:d
dx (−2x2) = −2·2x2−1=−4x
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate −1:d
dx (−1) = 0
Therefore, the derivative of the given function f(x)=4x3−2x2+
5x−1is:
f′(x) = 12x2−4x+ 5
Question 5
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.Question 5: Find the absolute maximum and absolute minimum
values of the function f(x) = x3−3x2+ 2 on the interval [0,2].
Step-by-step solution: 1. Find the critical points by taking the
derivative of f(x)and setting it to zero:
f′(x)=3x2−6x= 0
3x(x−2) = 0
x= 0 or x= 2
4
2. Check the endpoints of the interval: - At x= 0:f(0) = 2 - At
x= 2:f(2) = 2
3. Evaluate the function at the critical points: - At x= 0:f(0) = 2
- At x= 2:f(2) = 2
4. Compare all values to determine the absolute maximum and
minimum: - Absolute maximum value is 2 and occurs at x= 0 and
x= 2. - Absolute minimum value is 2 and also occurs at x= 0 and
x= 2.
Question 6
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x) = 0 to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.Question 6: Find the absolute maximum
and absolute minimum values of the function f(x) = x3−3x2+ 2 on
the interval [−1,3].
Step-by-step Solution: 1. Find the critical points of f(x)within
the interval [−1,3]:
f′(x)=3x2−6x
Setting f′(x)=0to find critical points:
3x2−6x= 0
3x(x−2) = 0
Critical points are x= 0 and x= 2.
5
2. Evaluate f(x)at the critical points and the endpoints of the
interval:
f(−1) = (−1)3−3(−1)2+ 2 = −2
f(0) = 03−3(0)2+ 2 = 2
f(2) = 23−3(2)2+ 2 = −6
f(3) = 33−3(3)2+ 2 = −11
3. Compare the values of f(x)at the critical points and endpoints:
-f(−1) = −2-f(0) = 2 -f(2) = −6-f(3) = −11
4. Therefore, the absolute maximum value of f(x)on the interval
[−1,3] is 2 and it occurs at x= 0, while the absolute minimum value
is -11 and it occurs at x= 3.
Question 7
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 7:
Find the derivative of the function f(x) = 3x2−2x+ 5.
Step-by-step solution:
Given function: f(x)=3x2−2x+ 5
To find the derivative of a function, we apply the power rule for
differentiation:
1. Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
6
2. Differentiate the term −2x:
d
dx (−2x) = −2
3. Differentiate the constant term 5:
d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x2−2x+ 5 is:
f′(x)=6x−2
Question 8
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.Certainly! Here’s a Calculus question along with
its step-by-step solution in LateX code:
Question 8: Find the derivative of the function f(x)=3x4−2x2+
5x−7.
Solution: To find the derivative of the function f(x) = 3x4−2x2+
5x−7, we apply the power rule of differentiation.
Given function: f(x)=3x4−2x2+ 5x−7
7
Step 1: Find the derivative of each term individually using the
power rule.
d
dx (3x4) = 4 ·3x4−1= 12x3
d
dx (−2x2)=2·2x2−1=−4x
d
dx (5x)=5
d
dx (−7) = 0
Step 2: Combine the derivatives of individual terms to find the
derivative of the function.
f′(x) = 12x3−4x+ 5
Therefore, the derivative of the function f(x)=3x4−2x2+ 5x−7is
f′(x) = 12x3−4x+ 5.
Question 9
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.Certainly! Here’s a Calculus question along
with the step-by-step solution in LateX code:
Question 9: Find the derivative of the following function: f(x) =
3x4−2x3+ 5x2
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2, we will apply the power rule for differentiation.
8
1. Find the derivative of xnwith respect to x:
d
dx xn=nxn−1
2. Apply the power rule to each term of the function:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)
= 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1
= 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2is
f′(x) = 12x3−6x2+ 10x.
Question 10
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a calculus question along
with its step-by-step solution in LateX code:
Question 10: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Solution: To find the derivative of the function, we differentiate
each term of the function with respect to x:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 11
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Sure, here is a question on Calculus for
Liberty University along with the step-by-step solutions in LaTeX
code:
Question 11: Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 9
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+9, we will differentiate each term separately using the power
rule:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule to each term:
f′(x)=4·3x4−1−3·2x3−1+ 2 ·5x2−1−7·1x1−1+ 0
Simplify the expression:
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 12
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
10
Therefore, the derivative of the function f(x) = e3xis 3e3x.Question
12: Find the derivative of the function f(x) = e3xusing the power rule
for derivatives.
Step-by-step solution: Given function: f(x) = e3x
To find the derivative, we will use the power rule for derivatives,
which states that the derivative of eax with respect to xis aeax.
Applying the power rule to our function f(x) = e3x, we have:
d
dx (e3x) = 3e3x
Therefore, the derivative of the function f(x) = e3xis 3e3x.
Question 13
Question 13:
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
apply the power rule and the constant multiple rule.
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.Sure, here is a calculus question along with its step-by-
step solution in LateX code:
Question 13:
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
apply the power rule and the constant multiple rule.
11
Given function: f(x)=3x2+ 4x−2
Step 1: Find the derivative of x2:
d
dx x2= 2x
Step 2: Find the derivative of 3x2using the constant multiple rule:
d
dx (3x2)=3 d
dx (x2) = 3(2x)=6x
Step 3: Find the derivative of 4xusing the power rule:
d
dx 4x= 4
Step 4: Combine the derivatives to find f′(x):
f′(x) = d
dx (3x2) + d
dx (4x)−d
dx (2) = 6x+ 4 −0=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−2is
f′(x) = 6x+ 4.
Question 14
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
12
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.Certainly! Here is a calculus question
along with its step-by-step solution in LateX code:
Question 14: Find the derivative of the following function using
the chain rule:
f(x) = (2x3+ 5x2+ 1)4
Solution: To find the derivative f′(x), we will use the chain rule.
Let’s denote the interior function as u= 2x3+ 5x2+ 1 and the exterior
function as g(u) = u4.
Step 1: Find du
dx by taking the derivative of the interior function:
du
dx =d
dx (2x3+ 5x2+ 1)
du
dx = 6x2+ 10x
Step 2: Find dg
du by taking the derivative of the exterior function
with respect to u:
dg
du =d
du (u4)
dg
du = 4u3
Step 3: Apply the chain rule to find the derivative of f(x):
f′(x) = dg
du ·du
dx
f′(x)=4u3·(6x2+ 10x)
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
Therefore, the derivative of the function f(x) = (2x3+ 5x2+1)4with
respect to xis:
f′(x) = 4(2x3+ 5x2+ 1)3·(6x2+ 10x)
This is the solution to question 14 regarding the derivative of a
function using the chain rule.
13
Question 15
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.Question 15: Find the derivative of the
function f(x) = 3x4+ 2x3−5x2+ 4x−1using the power rule.
Step-by-step solution: The power rule states that the derivative
of a term axnis given by n·a·xn−1.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
To find the derivative of f(x), we apply the power rule to each
term: f′(x) = d
dx [3x4] + d
dx [2x3]−d
dx [5x2] + d
dx [4x]−d
dx [1]
Now, using the power rule for differentiation: f′(x) = 3 ·4x4−1+ 2 ·
3x3−1−5·2x2−1+ 4 ·1x1−1−0
Simplify the derivatives: f′(x) = 12x3+ 6x2−10x+ 4
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is f′(x) = 12x3+ 6x2−10x+ 4.
Question 16
Step-by-step solution: To find the derivative of the function f(x) =
4x3−2x2+ 5x−7, we will differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
14
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 16: Find the derivative of 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 differentiate each term with respect to x
using the power rule for derivatives.
1. Differentiate the term 4x3:
d
dx (4x3) = 3 ·4x3−1= 12x2
2. Differentiate the term −2x2:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate the term 5x:
d
dx (5x)=5
4. Differentiate the constant term −7:
d
dx (−7) = 0
Therefore, the derivative of the function f(x)=4x3−2x2+ 5x−7is:
f′(x) = 12x2−4x+ 5
Question 17
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
15
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7. Question 17:
Find the derivative of the following function:
f(x)=3x4−2x2+ 7x−4
Solution:
To find the derivative of the function f(x) = 3x4−2x2+ 7x−4, we
will apply the power rule for differentiation.
1. Find the derivative of x4:
d
dx (x4) = 4x3
2. Find the derivative of x2:
d
dx (x2)=2x
3. Find the derivative of x:
d
dx (x) = 1
Now, use the linearity of the derivative to find the derivative of
the function f(x):
f′(x) = 3(4x3)−2(2x) + 7(1) −0
⇒f′(x) = 12x3−4x+ 7
Therefore, the derivative of the function f(x)=3x4−2x2+ 7x−4is
f′(x) = 12x3−4x+ 7.
Question 18
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
16
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.Certainly! Here’s a Calculus
question along with its step-by-step solution in LaTeX code:
“‘latex Question 18: Find the derivative of the function f(x) =
3x3−2x2+ 5x−7.
Solution: To find the derivative of the function f(x) = 3x3−2x2+
5x−7, we differentiate each term with respect to x.
The derivative of 3x3is:
d
dx (3x3)=3·3x3−1= 9x2
The derivative of −2x2is:
d
dx (−2x2) = −2·2x2−1=−4x
The derivative of 5xis:
d
dx (5x)=5
The derivative of −7(a constant term) is zero.
Therefore, the derivative of the function f(x)=3x3−2x2+ 5x−7is:
f′(x) = 9x2−4x+ 5
“‘
Feel free to use this code for your assignment at Liberty University!
Let me know if you need any more help.
17
Question 19
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.Question
19:
Find the derivative of the function f(x) = 3
x4.
Solution:
To find the derivative of the function f(x) = 3
x4, we need to use the
power rule for differentiation. The power rule states that if f(x) = xn,
then f′(x) = nxn−1.
Given the function f(x) = 3
x4, we can rewrite it as f(x)=3x−4.
Now, using the power rule, we find the derivative:
f′(x) = −4·3x−4−1=−12x−5=−12
x5
Therefore, the derivative of the function f(x) = 3
x4is f′(x) = −12
x5.
Question 20
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
to express the integrand as a sum of partial fractions: x2−3x+2
x(x−1) =
A
x+B
x−1. 5. Find the values of Aand Bby equating coefficients
and solving the resulting system of linear equations. 6. The partial
fraction decomposition is x2−3x+2
x(x−1) =1
x−2
x−1. 7. Integrate each partial
fraction separately to get R1
xdx −R2
x−1dx. 8. The antiderivative of
1
xis ln |x|. 9. The antiderivative of 2
x−1is 2 ln |x−1|. 10. Thus,
the indefinite integral of the original function is ln |x| − 2 ln |x−1|+C,
where Cis the constant of integration.Question 20: Find the indefinite
integral Rx2−3x+2
x2−xdx.
Step-by-step solution: 1. Write the given integral as Rx2−3x+2
x2−xdx.
2. Factor the denominator x2−xto get x(x−1). 3. Rewrite the
integral as Rx2−3x+2
x(x−1) dx. 4. Divide the numerator by the denominator
18
to express the integrand as a sum of partial fractions: x2−3x+2
x(x−1) =
A
x+B
x−1. 5. Find the values of Aand Bby equating coefficients
and solving the resulting system of linear equations. 6. The partial
fraction decomposition is x2−3x+2
x(x−1) =1
x−2
x−1. 7. Integrate each partial
fraction separately to get R1
xdx −R2
x−1dx. 8. The antiderivative of
1
xis ln |x|. 9. The antiderivative of 2
x−1is 2 ln |x−1|. 10. Thus, the
indefinite integral of the original function is ln |x|−2 ln |x−1|+C, where
Cis the constant of integration.
19
