MATH 100 - FUNDAMENTALS OF
MATHEMATICS - Calculus Question Bank - Set
1
Question 1
Question 1: Find the derivative of the function f(x)=3x2+ 4x−2 using
the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−2 using the power rule,
we differentiate each term individually.
1. Differentiate the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
2. Differentiate the term 4x:
d
dx (4x)=4·1x1−1= 4
3. Differentiate the constant term −2:
d
dx (−2) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−2 is:
f′(x)=6x+ 4
Certainly! Here is a calculus question for Liberty University in LateX
code:
Question 1: Find the derivative of the function f(x) = 3x2+ 4x−2
using the power rule.
Solution: To find the derivative of f(x)=3x2+ 4x−2using the
power rule, we differentiate each term individually.
1. Differentiate the term 3x2:
d
dx (3x2)=3·2x2−1= 6x
1
2. Differentiate the term 4x:
d
dx (4x)=4·1x1−1= 4
3. Differentiate the constant term −2:
d
dx (−2) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−2is:
f′(x)=6x+ 4
Question 2
Question 2: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will use the power rule of differentiation. The power rule
states that if f(x) = axn, then the derivative of f(x)with respect to x
is f′(x) = anx(n−1).
Given function: f(x) = 3x4+ 2x3−5x2+ 7 Applying the power rule:
f′(x) = 4(3)x4−1+ 3(2)x3−1−2(5)x2−1+ 0
f′(x) = 12x3+ 6x2−10x
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
I hope this solution helps! Let me know if you need help with
anything else.Sure, I can help with that! Here’s a question along
with its step-by-step solution in LateX code:
Question 2: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will use the power rule of differentiation. The power rule
states that if f(x) = axn, then the derivative of f(x)with respect to x
is f′(x) = anx(n−1).
Given function: f(x)=3x4+ 2x3−5x2+ 7 Applying the power rule:
f′(x) = 4(3)x4−1+ 3(2)x3−1−2(5)x2−1+ 0
f′(x) = 12x3+ 6x2−10x
2
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
I hope this solution helps! Let me know if you need help with
anything else.
Question 3
Find the derivative of the function f(x) = 3x4−8x2+ 5.
Step-by-step solution:
To find the derivative of the function f(x)=3x4−8x2+ 5, we will
differentiate each term with respect to xusing the power rule for
differentiation.
The power rule states that if we have a term of the form axn, the
derivative is n×axn−1.
1. Differentiate the term 3x4:
d
dx (3x4)=4×3x4−1= 12x3
2. Differentiate the term −8x2:
d
dx (−8x2) = 2 ×(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
f′(x) = 12x3−16x
Question 3:
Find the derivative of the function f(x) = 3x4−8x2+ 5.
Step-by-step solution:
To find the derivative of the function f(x)=3x4−8x2+ 5, we will
differentiate each term with respect to xusing the power rule for
differentiation.
The power rule states that if we have a term of the form axn, the
derivative is n×axn−1.
1. Differentiate the term 3x4:
d
dx (3x4)=4×3x4−1= 12x3
2. Differentiate the term −8x2:
d
dx (−8x2) = 2 ×(−8)x2−1=−16x
3
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
f′(x) = 12x3−16x
Question 4
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+1
To find the derivative of the function, we will differentiate each
term individually using the power rule of derivatives.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. Differentiate the constant term 1: The derivative of a constant
term is always zero.
Therefore, the derivative of the given function f(x)=3x4−2x3+
5x2−7x+ 1 is:
f′(x) = 12x3−6x2+ 10x−7
Question 4: Find the derivative of the function f(x)=3x4−2x3+ 5x2−
7x+ 1.
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+1
To find the derivative of the function, we will differentiate each
term individually using the power rule of derivatives.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
4
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. Differentiate the constant term 1: The derivative of a constant
term is always zero.
Therefore, the derivative of the given function f(x)=3x4−2x3+
5x2−7x+ 1 is:
f′(x) = 12x3−6x2+ 10x−7
Question 5
Question 5: Find the derivative of the function f(x) = 3x2+ 5x−2.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x−2, we will use the power rule for derivatives.
Given function: f(x)=3x2+ 5x−2
Using the power rule, the derivative of xnis nxn−1.
Therefore, the derivative of the function is:
f′(x) = d
dx (3x2) + d
dx (5x)−d
dx (2)
f′(x)=3·2x2−1+ 5 ·1x1−1−0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x−2is
f′(x) = 6x+ 5.Sure, here is a calculus question along with a step-by-
step solution in LaTeX code:
Question 5: Find the derivative of the function f(x) = 3x2+ 5x−2.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x−2, we will use the power rule for derivatives.
Given function: f(x)=3x2+ 5x−2
Using the power rule, the derivative of xnis nxn−1.
Therefore, the derivative of the function is:
f′(x) = d
dx (3x2) + d
dx (5x)−d
dx (2)
f′(x)=3·2x2−1+ 5 ·1x1−1−0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x−2is
f′(x) = 6x+ 5.
5
Question 6
Step-by-step Solution: To find the derivative of the function f(x) =
3x4−8x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −8x2:
d
dx (−8x2)=2·(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is 0.
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
d
dx (f(x)) = 12x3−16x
Question 6: Find the derivative of the function f(x)=3x4−8x2+ 5.
Step-by-step Solution: To find the derivative of the function f(x) =
3x4−8x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −8x2:
d
dx (−8x2)=2·(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is 0.
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
d
dx (f(x)) = 12x3−16x
Question 7
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−1, we will use the power rule, which states that the derivative of
xnwith respect to xis nxn−1.
f′(x) = d
dx (5x3)−d
dx (2x2) + d
dx (7x)−d
dx (1)
f′(x) = 15x2−4x+ 7
6
Therefore, the derivative of the function f(x)=5x3−2x2+ 7x−1is
f′(x) = 15x2−4x+ 7.Question 7: Find the derivative of the following
function using the power rule:
f(x)=5x3−2x2+ 7x−1
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−1, we will use the power rule, which states that the derivative of
xnwith respect to xis nxn−1.
f′(x) = d
dx (5x3)−d
dx (2x2) + d
dx (7x)−d
dx (1)
f′(x) = 15x2−4x+ 7
Therefore, the derivative of the function f(x)=5x3−2x2+ 7x−1is
f′(x) = 15x2−4x+ 7.
Question 8
Question 8: Find the derivative of the function f(x) = 3x2−5x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−5x+ 2,
we will use the power rule of differentiation. The power rule states
that the derivative of xnis nxn−1.
Given function: f(x)=3x2−5x+ 2
Step 1: Find the derivative of 3x2.
d
dx (3x2) = 2 ×3x2−1
= 6x
Step 2: Find the derivative of −5x.
d
dx (−5x) = −5×1x1−1
=−5
Step 3: Find the derivative of the constant term 2. Since the
derivative of a constant term is 0, the derivative of 2is 0.
Step 4: Combine the derivatives of each term to find the derivative
of the function.
f′(x)=6x−5+0
= 6x−5
Therefore, the derivative of the function f(x) = 3x2−5x+ 2 is
f′(x) = 6x−5.Sure! Here is a Calculus question along with its step-
by-step solution in LaTeX code:
Question 8: Find the derivative of the function f(x) = 3x2−5x+ 2.
7
Solution: To find the derivative of the function f(x) = 3x2−5x+ 2,
we will use the power rule of differentiation. The power rule states
that the derivative of xnis nxn−1.
Given function: f(x)=3x2−5x+ 2
Step 1: Find the derivative of 3x2.
d
dx (3x2) = 2 ×3x2−1
= 6x
Step 2: Find the derivative of −5x.
d
dx (−5x) = −5×1x1−1
=−5
Step 3: Find the derivative of the constant term 2. Since the
derivative of a constant term is 0, the derivative of 2is 0.
Step 4: Combine the derivatives of each term to find the derivative
of the function.
f′(x)=6x−5+0
= 6x−5
Therefore, the derivative of the function f(x) = 3x2−5x+ 2 is
f′(x) = 6x−5.
Question 9
Find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1.
Step-by-step solution:
To find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1,
we will use the power rule of differentiation.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
Taking derivatives term by term:
1. d
dx (3x4)=4·3x4−1= 12x32. d
dx (2x3) = 3 ·2x3−1= 6x23. d
dx (−5x2) =
2·(−5)x2−1=−10x4. d
dx (4x)=1·4x1−1= 4 5. d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is:
f′(x) = 12x3+ 6x2−10x+ 4
Question 9:
Find the derivative of the function f(x) = 3x4+ 2x3−5x2+ 4x−1.
Step-by-step solution:
To find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1,
we will use the power rule of differentiation.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
8
Taking derivatives term by term:
1. d
dx (3x4)=4·3x4−1= 12x32. d
dx (2x3) = 3 ·2x3−1= 6x23. d
dx (−5x2) =
2·(−5)x2−1=−10x4. d
dx (4x)=1·4x1−1= 4 5. d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is:
f′(x) = 12x3+ 6x2−10x+ 4
Question 10
Question 10: Find the derivative of the function f(x)=3x2−4x+ 6.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 6,
we will differentiate each term with respect to x using the power rule
for derivatives.
Step 1: Differentiate the term 3x2:
d
dx (3x2)=3·2x= 6x
Step 2: Differentiate the term −4x:
d
dx (−4x) = −4
Step 3: Differentiate the constant term 6:
d
dx 6=0
Therefore, the derivative of the function f(x)=3x2−4x+ 6 is:
df
dx = 6x−4
Sure! Here is a Calculus question along with the step-by-step solution
in LateX code:
Question 10: Find the derivative of the function f(x)=3x2−4x+ 6.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 6,
we will differentiate each term with respect to x using the power rule
for derivatives.
Step 1: Differentiate the term 3x2:
d
dx (3x2)=3·2x= 6x
Step 2: Differentiate the term −4x:
d
dx (−4x) = −4
9
Step 3: Differentiate the constant term 6:
d
dx 6=0
Therefore, the derivative of the function f(x)=3x2−4x+ 6 is:
df
dx = 6x−4
Question 11
Find the derivative of the following function: f(x)=3x2−4x+ 5.
Solution:
To find the derivative of a function, we apply the power rule which
states that the derivative of xnwith respect to xis nxn−1, where nis
a constant.
Given function: f(x)=3x2−4x+ 5
To find the derivative, we differentiate each term of the function
separately:
1. Differentiate 3x2:
d
dx (3x2)=3·2x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 5: The derivative of a constant
is always 0.
Therefore, the derivative of the given function f(x)=3x2−4x+ 5
is:
f′(x)=6x−4
“‘“‘latex Question 11:
Find the derivative of the following function: f(x)=3x2−4x+ 5.
Solution:
To find the derivative of a function, we apply the power rule which
states that the derivative of xnwith respect to xis nxn−1, where nis
a constant.
Given function: f(x)=3x2−4x+ 5
To find the derivative, we differentiate each term of the function
separately:
1. Differentiate 3x2:
d
dx (3x2)=3·2x2−1= 6x
10
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 5: The derivative of a constant
is always 0.
Therefore, the derivative of the given function f(x)=3x2−4x+ 5
is:
f′(x)=6x−4
“‘
Question 12
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x−1, we will differentiate each term with respect to x
using the power rule.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate the constant term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−1is:
f′(x) = 12x3−6x2+ 5
Question 12: Find the derivative of the function f(x) = 3x4−2x3+5x−1.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x−1, we will differentiate each term with respect to x
using the power rule.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
11
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate the constant term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−1is:
f′(x) = 12x3−6x2+ 5
Question 13
Question 13: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Step-by-step solution:
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of f(x), we differentiate each term with re-
spect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7+0
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Certainly! Here is a Calculus question
along with the step-by-step solution in LateX code:
Question 13: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Step-by-step solution:
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of f(x), we differentiate each term with re-
spect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7+0
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
12
Question 14
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+9
To find the derivative f′(x), we differentiate each term of the func-
tion with respect to xusing the power rule of differentiation.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3) = 3 · −2x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1· −7x1−1=−7
5. Differentiate the constant term 9: Since the derivative of a
constant is 0, the derivative of the constant term 9is 0.
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 14: Find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9.
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+9
To find the derivative f′(x), we differentiate each term of the func-
tion with respect to xusing the power rule of differentiation.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3) = 3 · −2x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1· −7x1−1=−7
13
5. Differentiate the constant term 9: Since the derivative of a
constant is 0, the derivative of the constant term 9is 0.
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 15
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 9, we will apply the power rule of differentiation:
1. Calculate the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Calculate the derivative of the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Calculate the derivative of the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Calculate the derivative of the term −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term 9is zero.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 15: Find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 9, we will apply the power rule of differentiation:
1. Calculate the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Calculate the derivative of the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Calculate the derivative of the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
14
4. Calculate the derivative of the term −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term 9is zero.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 16
Step-by-step solution:
f(x) = 3x4−2x3+ 5x2−7x+ 9
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Question 16: Find the derivative of the
function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Step-by-step solution:
f(x) = 3x4−2x3+ 5x2−7x+ 9
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 17
Question 17: 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 will differentiate each term with respect to
xusing the power rule for derivatives:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
15
Now, applying the power rule:
f′(x)=3·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
Simplifying the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Sure! Here is a question on Calculus:
Question 17: 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 will differentiate each term with respect to
xusing the power rule for derivatives:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule:
f′(x)=3·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
Simplifying the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 18
Question 18: Find the derivative of the function f(x)=3x2+ 5x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 5x−2,
we will apply the power rule for derivatives.
Step 1: Find the derivative of 3x2.
f′(x) = 3 ·2x2−1= 6x
Step 2: Find the derivative of 5x.
f′(x) = 5 ·1x1−1= 5
16
Step 3: Find the derivative of the constant term −2.
f′(x)=0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Feel free to reach out if you need further assistance or more ques-
tions!Sure! Here is a Calculus question along with its step-by-step
solution in LateX code:
Question 18: Find the derivative of the function f(x)=3x2+ 5x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 5x−2,
we will apply the power rule for derivatives.
Step 1: Find the derivative of 3x2.
f′(x)=3·2x2−1= 6x
Step 2: Find the derivative of 5x.
f′(x) = 5 ·1x1−1= 5
Step 3: Find the derivative of the constant term −2.
f′(x)=0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Feel free to reach out if you need further assistance or more ques-
tions!
Question 19
Question 19: Find the derivative of the function f(x)=3x2+ 4x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 4x−2,
we can use the power rule for differentiation.
Step 1: Find the derivative of the term 3x2.
d
dx (3x2)=2×3×x2−1= 6x
Step 2: Find the derivative of the term 4x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −2.
d
dx (−2) = 0
17
Step 4: Combine the derivatives of individual terms to find the
derivative of the function f(x) = 3x2+ 4x−2.
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.
This completes the solution.Sure! Here is a Calculus question
along with its step-by-step solution in LateX code:
Question 19: Find the derivative of the function f(x)=3x2+ 4x−2.
Solution: To find the derivative of the function f(x)=3x2+ 4x−2,
we can use the power rule for differentiation.
Step 1: Find the derivative of the term 3x2.
d
dx (3x2)=2×3×x2−1= 6x
Step 2: Find the derivative of the term 4x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −2.
d
dx (−2) = 0
Step 4: Combine the derivatives of individual terms to find the
derivative of the function f(x) = 3x2+ 4x−2.
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.
This completes the solution.
Question 20
f(x)=3x4−8x3+ 2x2−5x+ 1
Step-by-step solution: Let’s differentiate using the power rule for
derivatives, which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−8x3+ 2x2−5x+ 1
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
18
2. Find the derivative of −8x3:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Find the derivative of 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. Find the derivative of −5x:
d
dx (−5x)=1·(−5)x1−1=−5
5. The derivative of a constant is 0:
d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−8x3+2x2−5x+1
is:
f′(x) = 12x3−24x2+ 4x−5
Question 20: Calculate the derivative of the following function:
f(x)=3x4−8x3+ 2x2−5x+ 1
Step-by-step solution: Let’s differentiate using the power rule for
derivatives, which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−8x3+ 2x2−5x+ 1
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x3:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Find the derivative of 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. Find the derivative of −5x:
d
dx (−5x)=1·(−5)x1−1=−5
5. The derivative of a constant is 0:
d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−8x3+2x2−5x+1
is:
f′(x) = 12x3−24x2+ 4x−5
19
Question 21
Question 21: Find the derivative of the function f(x)=2x3−5x2+
4x−7.
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule of
differentiation.
f′(x) = d
dx (2x3)−d
dx (5x2) + d
dx (4x)−d
dx (7)
= 2 ·3x3−1−5·2x2−1+ 4 ·1x1−1−0
= 6x2−10x+ 4
Therefore, the derivative of the function f(x)=2x3−5x2+ 4x−7is
f′(x) = 6x2−10x+ 4.Certainly! Here is a Calculus question along with
its step-by-step solution in LateX code:
Question 21: Find the derivative of the function f(x)=2x3−5x2+
4x−7.
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule of
differentiation.
f′(x) = d
dx (2x3)−d
dx (5x2) + d
dx (4x)−d
dx (7)
= 2 ·3x3−1−5·2x2−1+ 4 ·1x1−1−0
= 6x2−10x+ 4
Therefore, the derivative of the function f(x)=2x3−5x2+ 4x−7is
f′(x) = 6x2−10x+ 4.
Question 22
Question 22:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 4
Step-by-step solution:
To find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 4,
we need to differentiate each term separately using the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (4)
Now, let’s calculate the derivatives of each term:
f′(x) = 12x3−6x2+ 10x−7
20
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is f′(x) = 12x3−6x2+ 10x−7.
This is the solution to question 22.Certainly! Here is a Calculus
question along with its step-by-step solution in LateX code:
Question 22:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 4
Step-by-step solution:
To find the derivative of the function f(x)=3x4−2x3+ 5x2−7x+ 4,
we need to differentiate each term separately using the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (4)
Now, let’s calculate the derivatives of each term:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is f′(x) = 12x3−6x2+ 10x−7.
This is the solution to question 22.
Question 23
Question 23: Find the derivative of the function f(x)=3x4−2x3+
5x2−7.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we will apply the power rule for differentiation.
Step 1: Calculate the derivative by applying the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7)
Step 2: Apply the power rule to find the derivatives.
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−0
f′(x) = 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.Sure! Here’s a Calculus question along with
the step-by-step solution in LateX code:
Question 23: Find the derivative of the function f(x)=3x4−2x3+
5x2−7.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we will apply the power rule for differentiation.
21
Step 1: Calculate the derivative by applying the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7)
Step 2: Apply the power rule to find the derivatives.
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−0
f′(x) = 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.
Question 24
Solution:
To find the derivative of the function f(x)=3x2−4x+ 5, we will
use the power rule for differentiation.
Given function, f(x)=3x2−4x+ 5
Now, differentiate each term of the function with respect to xusing
the power rule:
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=2·3x2−1−4·1x1−1+ 0
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.24. Find the derivative of the function f(x)=3x2−4x+5.
Solution:
To find the derivative of the function f(x)=3x2−4x+ 5, we will
use the power rule for differentiation.
Given function, f(x)=3x2−4x+ 5
Now, differentiate each term of the function with respect to xusing
the power rule:
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=2·3x2−1−4·1x1−1+ 0
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.
22
Question 25
Find the derivative of the function f(x) = 5x3−2x2+ 4x−7.
Step-by-step solution:
To find the derivative of the function f(x) = 5x3−2x2+ 4x−7, we
need to differentiate each term with respect to x.
1. Differentiate 5x3with respect to x:
d
dx (5x3) = 3 ·5x3−1= 15x2
2. Differentiate −2x2with respect to x:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate 4xwith respect to x:
d
dx (4x)=4
4. Differentiate the constant term −7with respect to x: Since −7
is a constant, its derivative is 0.
Therefore, the derivative of the function f(x)=5x3−2x2+ 4x−7is:
f′(x) = 15x2−4x+ 4
So, the derivative of the function is f′(x) = 15x2−4x+ 4.Question
25:
Find the derivative of the function f(x) = 5x3−2x2+ 4x−7.
Step-by-step solution:
To find the derivative of the function f(x) = 5x3−2x2+ 4x−7, we
need to differentiate each term with respect to x.
1. Differentiate 5x3with respect to x:
d
dx (5x3) = 3 ·5x3−1= 15x2
2. Differentiate −2x2with respect to x:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate 4xwith respect to x:
d
dx (4x)=4
4. Differentiate the constant term −7with respect to x: Since −7
is a constant, its derivative is 0.
Therefore, the derivative of the function f(x)=5x3−2x2+ 4x−7is:
f′(x) = 15x2−4x+ 4
So, the derivative of the function is f′(x) = 15x2−4x+ 4.
23
Question 26
Question 26: Find the derivative of the following function: f(x) =
3x4+ 2x3−5x2+ 8.
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+8, we will differentiate each term separately using the power rule
for derivatives.
Given function: f(x)=3x4+ 2x3−5x2+ 8.
The derivative of 3x4is obtained by multiplying the coefficient by
the power and then reducing the power by 1.
d
dx (3x4)=4·3x4−1= 12x3.
The derivative of 2x3is calculated in a similar manner using the
power rule.
d
dx (2x3)=3·2x3−1= 6x2.
Similarly, the derivative of −5x2is found as follows:
d
dx (−5x2)=2·(−5)x2−1=−10x.
The derivative of a constant term like 8 is always 0.
d
dx (8) = 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 8
is:
f′(x) = 12x3+ 6x2−10x.
Thus, the derivative of the function is f′(x) = 12x3+6x2−10x.Certainly!
Here’s a Calculus question along with its step-by-step solution in La-
teX code:
Question 26: Find the derivative of the following function: f(x) =
3x4+ 2x3−5x2+ 8.
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+8, we will differentiate each term separately using the power rule
for derivatives.
Given function: f(x)=3x4+ 2x3−5x2+ 8.
The derivative of 3x4is obtained by multiplying the coefficient by
the power and then reducing the power by 1.
d
dx (3x4)=4·3x4−1= 12x3.
The derivative of 2x3is calculated in a similar manner using the
power rule.
d
dx (2x3)=3·2x3−1= 6x2.
Similarly, the derivative of −5x2is found as follows:
d
dx (−5x2)=2·(−5)x2−1=−10x.
The derivative of a constant term like 8 is always 0.
d
dx (8) = 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 8
is:
f′(x) = 12x3+ 6x2−10x.
Thus, the derivative of the function is f′(x) = 12x3+ 6x2−10x.
24
Question 27
Question 27: Find the derivative of the function f(x)=3x2+ 5x+ 8.
Solution: To find the derivative of the function f(x) = 3x2+ 5x+ 8,
we use the power rule of differentiation.
Step 1: Identify the function The function given is f(x)=3x2+
5x+ 8.
Step 2: Apply the power rule The power rule states that if f(x) =
axn, then f′(x) = n·axn−1. Using this rule, we differentiate each term
of the function separately.
Step 3: Differentiate each term - Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
- Differentiate the term 5x:
d
dx (5x)=1·5x1−1= 5
- Differentiate the constant term 8: The derivative of a constant is 0.
Step 4: Combine the derivatives Putting together the derivatives
of each term, we get the derivative of the function:
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 8 is
f′(x) = 6x+ 5.Sure! Here is a Calculus question along with its step-
by-step solution in LateX code:
Question 27: Find the derivative of the function f(x)=3x2+ 5x+ 8.
Solution: To find the derivative of the function f(x) = 3x2+ 5x+ 8,
we use the power rule of differentiation.
Step 1: Identify the function The function given is f(x)=3x2+
5x+ 8.
Step 2: Apply the power rule The power rule states that if f(x) =
axn, then f′(x) = n·axn−1. Using this rule, we differentiate each term
of the function separately.
Step 3: Differentiate each term - Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
- Differentiate the term 5x:
d
dx (5x)=1·5x1−1= 5
- Differentiate the constant term 8: The derivative of a constant is 0.
Step 4: Combine the derivatives Putting together the derivatives
of each term, we get the derivative of the function:
f′(x)=6x+ 5
25
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 8 is
f′(x) = 6x+ 5.
Question 28
Question 28: Find the derivative of the function f(x)=3x2+ 4x−7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 4x−7, we need to differentiate each term with respect to xusing
the power rule.
Given function: f(x)=3x2+ 4x−7
Step 1: Differentiate the term 3x2:
f′(x) = d
dx (3x2)
f′(x)=2·3x2−1
f′(x)=6x
Step 2: Differentiate the term 4x:
f′(x) = d
dx (4x)
f′(x) = 4 ·1
f′(x)=4
Step 3: Differentiate the constant term −7: The derivative of a
constant is zero, so it does not contribute to the derivative of the
function.
Step 4: Combine the derivatives of each term to find the derivative
of the function f(x) = 3x2+ 4x−7:
f′(x)=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−7is
f′(x)=6x+ 4.Certainly! Here’s a Calculus question along with step-
by-step solution in LateX code:
Question 28: Find the derivative of the function f(x)=3x2+ 4x−7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 4x−7, we need to differentiate each term with respect to xusing
the power rule.
Given function: f(x)=3x2+ 4x−7
Step 1: Differentiate the term 3x2:
f′(x) = d
dx (3x2)
f′(x)=2·3x2−1
26
f′(x)=6x
Step 2: Differentiate the term 4x:
f′(x) = d
dx (4x)
f′(x) = 4 ·1
f′(x)=4
Step 3: Differentiate the constant term −7: The derivative of a
constant is zero, so it does not contribute to the derivative of the
function.
Step 4: Combine the derivatives of each term to find the derivative
of the function f(x) = 3x2+ 4x−7:
f′(x)=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−7is
f′(x) = 6x+ 4.
Question 29
Question 29: Find the derivative of the following function: f(x) =
3x2+ 4x−5
Solution:
To find the derivative of f(x)=3x2+4x−5, we will apply the power
rule for derivatives and the constant multiple rule.
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4x:
d
dx (4x)=4
Step 3: Combine the derivatives of individual terms:
f′(x)=6x+ 4
Therefore, the derivative of f(x)=3x2+ 4x−5is f′(x)=6x+
4.Certainly! Here is a calculus question along with its step-by-step
solution written in LateX code:
Question 29: Find the derivative of the following function: f(x) =
3x2+ 4x−5
Solution:
To find the derivative of f(x)=3x2+4x−5, we will apply the power
rule for derivatives and the constant multiple rule.
27
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4x:
d
dx (4x)=4
Step 3: Combine the derivatives of individual terms:
f′(x)=6x+ 4
Therefore, the derivative of f(x)=3x2+ 4x−5is f′(x)=6x+ 4.
Question 30
Question 30: Find the derivative of the function f(x)=3x2+ 4x−5
using the power rule.
Solution: To find the derivative of the function using the power
rule, we differentiate each term separately. The power rule states
that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x2+ 4x−5
Differentiate each term:
d
dx (3x2)=2·3x2−1= 6x
d
dx (4x)=1·4x1−1= 4
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
This is the final answer.
f′(x)=6x+ 4
Certainly! Here is a calculus question along with the step-by-step
solution in LateX code:
Question 30: Find the derivative of the function f(x)=3x2+ 4x−5
using the power rule.
Solution: To find the derivative of the function using the power
rule, we differentiate each term separately. The power rule states
that if f(x) = axn, then f′(x) = anxn−1.
28
2. Differentiate the term 4x:
d
dx (4x)=4·1x1−1= 4
3. Differentiate the constant term −2:
d
dx (−2) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−2is:
f′(x)=6x+ 4
Question 2
Question 2: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will use the power rule of differentiation. The power rule
states that if f(x) = axn, then the derivative of f(x)with respect to x
is f′(x) = anx(n−1).
Given function: f(x) = 3x4+ 2x3−5x2+ 7 Applying the power rule:
f′(x) = 4(3)x4−1+ 3(2)x3−1−2(5)x2−1+ 0
f′(x) = 12x3+ 6x2−10x
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
I hope this solution helps! Let me know if you need help with
anything else.Sure, I can help with that! Here’s a question along
with its step-by-step solution in LateX code:
Question 2: Find the derivative of the following function:
f(x) = 3x4+ 2x3−5x2+ 7
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+ 7, we will use the power rule of differentiation. The power rule
states that if f(x) = axn, then the derivative of f(x)with respect to x
is f′(x) = anx(n−1).
Given function: f(x)=3x4+ 2x3−5x2+ 7 Applying the power rule:
f′(x) = 4(3)x4−1+ 3(2)x3−1−2(5)x2−1+ 0
f′(x) = 12x3+ 6x2−10x
2
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 7
is:
f′(x) = 12x3+ 6x2−10x
I hope this solution helps! Let me know if you need help with
anything else.
Question 3
Find the derivative of the function f(x) = 3x4−8x2+ 5.
Step-by-step solution:
To find the derivative of the function f(x)=3x4−8x2+ 5, we will
differentiate each term with respect to xusing the power rule for
differentiation.
The power rule states that if we have a term of the form axn, the
derivative is n×axn−1.
1. Differentiate the term 3x4:
d
dx (3x4)=4×3x4−1= 12x3
2. Differentiate the term −8x2:
d
dx (−8x2) = 2 ×(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
f′(x) = 12x3−16x
Question 3:
Find the derivative of the function f(x) = 3x4−8x2+ 5.
Step-by-step solution:
To find the derivative of the function f(x)=3x4−8x2+ 5, we will
differentiate each term with respect to xusing the power rule for
differentiation.
The power rule states that if we have a term of the form axn, the
derivative is n×axn−1.
1. Differentiate the term 3x4:
d
dx (3x4)=4×3x4−1= 12x3
2. Differentiate the term −8x2:
d
dx (−8x2) = 2 ×(−8)x2−1=−16x
3
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is: d
dx (5) = 0
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
f′(x) = 12x3−16x
Question 4
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+1
To find the derivative of the function, we will differentiate each
term individually using the power rule of derivatives.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. Differentiate the constant term 1: The derivative of a constant
term is always zero.
Therefore, the derivative of the given function f(x)=3x4−2x3+
5x2−7x+ 1 is:
f′(x) = 12x3−6x2+ 10x−7
Question 4: Find the derivative of the function f(x)=3x4−2x3+ 5x2−
7x+ 1.
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+1
To find the derivative of the function, we will differentiate each
term individually using the power rule of derivatives.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
4
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. Differentiate the constant term 1: The derivative of a constant
term is always zero.
Therefore, the derivative of the given function f(x)=3x4−2x3+
5x2−7x+ 1 is:
f′(x) = 12x3−6x2+ 10x−7
Question 5
Question 5: Find the derivative of the function f(x) = 3x2+ 5x−2.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x−2, we will use the power rule for derivatives.
Given function: f(x)=3x2+ 5x−2
Using the power rule, the derivative of xnis nxn−1.
Therefore, the derivative of the function is:
f′(x) = d
dx (3x2) + d
dx (5x)−d
dx (2)
f′(x)=3·2x2−1+ 5 ·1x1−1−0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x−2is
f′(x) = 6x+ 5.Sure, here is a calculus question along with a step-by-
step solution in LaTeX code:
Question 5: Find the derivative of the function f(x) = 3x2+ 5x−2.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 5x−2, we will use the power rule for derivatives.
Given function: f(x)=3x2+ 5x−2
Using the power rule, the derivative of xnis nxn−1.
Therefore, the derivative of the function is:
f′(x) = d
dx (3x2) + d
dx (5x)−d
dx (2)
f′(x)=3·2x2−1+ 5 ·1x1−1−0
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x−2is
f′(x) = 6x+ 5.
5
Question 6
Step-by-step Solution: To find the derivative of the function f(x) =
3x4−8x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −8x2:
d
dx (−8x2)=2·(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is 0.
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
d
dx (f(x)) = 12x3−16x
Question 6: Find the derivative of the function f(x)=3x4−8x2+ 5.
Step-by-step Solution: To find the derivative of the function f(x) =
3x4−8x2+ 5, we will use the power rule for differentiation.
1. Find the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of the term −8x2:
d
dx (−8x2)=2·(−8)x2−1=−16x
3. Since the derivative of a constant is 0, the derivative of the
constant term 5is 0.
Therefore, the derivative of the function f(x)=3x4−8x2+ 5 is:
d
dx (f(x)) = 12x3−16x
Question 7
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−1, we will use the power rule, which states that the derivative of
xnwith respect to xis nxn−1.
f′(x) = d
dx (5x3)−d
dx (2x2) + d
dx (7x)−d
dx (1)
f′(x) = 15x2−4x+ 7
6
Therefore, the derivative of the function f(x)=5x3−2x2+ 7x−1is
f′(x) = 15x2−4x+ 7.Question 7: Find the derivative of the following
function using the power rule:
f(x)=5x3−2x2+ 7x−1
Solution: To find the derivative of the function f(x) = 5x3−2x2+
7x−1, we will use the power rule, which states that the derivative of
xnwith respect to xis nxn−1.
f′(x) = d
dx (5x3)−d
dx (2x2) + d
dx (7x)−d
dx (1)
f′(x) = 15x2−4x+ 7
Therefore, the derivative of the function f(x)=5x3−2x2+ 7x−1is
f′(x) = 15x2−4x+ 7.
Question 8
Question 8: Find the derivative of the function f(x) = 3x2−5x+ 2.
Solution: To find the derivative of the function f(x) = 3x2−5x+ 2,
we will use the power rule of differentiation. The power rule states
that the derivative of xnis nxn−1.
Given function: f(x)=3x2−5x+ 2
Step 1: Find the derivative of 3x2.
d
dx (3x2) = 2 ×3x2−1
= 6x
Step 2: Find the derivative of −5x.
d
dx (−5x) = −5×1x1−1
=−5
Step 3: Find the derivative of the constant term 2. Since the
derivative of a constant term is 0, the derivative of 2is 0.
Step 4: Combine the derivatives of each term to find the derivative
of the function.
f′(x)=6x−5+0
= 6x−5
Therefore, the derivative of the function f(x) = 3x2−5x+ 2 is
f′(x) = 6x−5.Sure! Here is a Calculus question along with its step-
by-step solution in LaTeX code:
Question 8: Find the derivative of the function f(x) = 3x2−5x+ 2.
7
Solution: To find the derivative of the function f(x) = 3x2−5x+ 2,
we will use the power rule of differentiation. The power rule states
that the derivative of xnis nxn−1.
Given function: f(x)=3x2−5x+ 2
Step 1: Find the derivative of 3x2.
d
dx (3x2) = 2 ×3x2−1
= 6x
Step 2: Find the derivative of −5x.
d
dx (−5x) = −5×1x1−1
=−5
Step 3: Find the derivative of the constant term 2. Since the
derivative of a constant term is 0, the derivative of 2is 0.
Step 4: Combine the derivatives of each term to find the derivative
of the function.
f′(x)=6x−5+0
= 6x−5
Therefore, the derivative of the function f(x) = 3x2−5x+ 2 is
f′(x) = 6x−5.
Question 9
Find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1.
Step-by-step solution:
To find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1,
we will use the power rule of differentiation.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
Taking derivatives term by term:
1. d
dx (3x4)=4·3x4−1= 12x32. d
dx (2x3) = 3 ·2x3−1= 6x23. d
dx (−5x2) =
2·(−5)x2−1=−10x4. d
dx (4x)=1·4x1−1= 4 5. d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is:
f′(x) = 12x3+ 6x2−10x+ 4
Question 9:
Find the derivative of the function f(x) = 3x4+ 2x3−5x2+ 4x−1.
Step-by-step solution:
To find the derivative of the function f(x)=3x4+ 2x3−5x2+ 4x−1,
we will use the power rule of differentiation.
Given function: f(x)=3x4+ 2x3−5x2+ 4x−1
8
Taking derivatives term by term:
1. d
dx (3x4)=4·3x4−1= 12x32. d
dx (2x3) = 3 ·2x3−1= 6x23. d
dx (−5x2) =
2·(−5)x2−1=−10x4. d
dx (4x)=1·4x1−1= 4 5. d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4+2x3−5x2+4x−1
is:
f′(x) = 12x3+ 6x2−10x+ 4
Question 10
Question 10: Find the derivative of the function f(x)=3x2−4x+ 6.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 6,
we will differentiate each term with respect to x using the power rule
for derivatives.
Step 1: Differentiate the term 3x2:
d
dx (3x2)=3·2x= 6x
Step 2: Differentiate the term −4x:
d
dx (−4x) = −4
Step 3: Differentiate the constant term 6:
d
dx 6=0
Therefore, the derivative of the function f(x)=3x2−4x+ 6 is:
df
dx = 6x−4
Sure! Here is a Calculus question along with the step-by-step solution
in LateX code:
Question 10: Find the derivative of the function f(x)=3x2−4x+ 6.
Solution: To find the derivative of the function f(x) = 3x2−4x+ 6,
we will differentiate each term with respect to x using the power rule
for derivatives.
Step 1: Differentiate the term 3x2:
d
dx (3x2)=3·2x= 6x
Step 2: Differentiate the term −4x:
d
dx (−4x) = −4
9
Step 3: Differentiate the constant term 6:
d
dx 6=0
Therefore, the derivative of the function f(x)=3x2−4x+ 6 is:
df
dx = 6x−4
Question 11
Find the derivative of the following function: f(x)=3x2−4x+ 5.
Solution:
To find the derivative of a function, we apply the power rule which
states that the derivative of xnwith respect to xis nxn−1, where nis
a constant.
Given function: f(x)=3x2−4x+ 5
To find the derivative, we differentiate each term of the function
separately:
1. Differentiate 3x2:
d
dx (3x2)=3·2x2−1= 6x
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 5: The derivative of a constant
is always 0.
Therefore, the derivative of the given function f(x)=3x2−4x+ 5
is:
f′(x)=6x−4
“‘“‘latex Question 11:
Find the derivative of the following function: f(x)=3x2−4x+ 5.
Solution:
To find the derivative of a function, we apply the power rule which
states that the derivative of xnwith respect to xis nxn−1, where nis
a constant.
Given function: f(x)=3x2−4x+ 5
To find the derivative, we differentiate each term of the function
separately:
1. Differentiate 3x2:
d
dx (3x2)=3·2x2−1= 6x
10
2. Differentiate −4x:
d
dx (−4x) = −4
3. Differentiate the constant term 5: The derivative of a constant
is always 0.
Therefore, the derivative of the given function f(x)=3x2−4x+ 5
is:
f′(x)=6x−4
“‘
Question 12
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x−1, we will differentiate each term with respect to x
using the power rule.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate the constant term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−1is:
f′(x) = 12x3−6x2+ 5
Question 12: Find the derivative of the function f(x) = 3x4−2x3+5x−1.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x−1, we will differentiate each term with respect to x
using the power rule.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
11
3. Differentiate 5x:d
dx (5x)=5
4. Differentiate the constant term −1:
d
dx (−1) = 0
Therefore, the derivative of the function f(x)=3x4−2x3+ 5x−1is:
f′(x) = 12x3−6x2+ 5
Question 13
Question 13: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Step-by-step solution:
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of f(x), we differentiate each term with re-
spect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7+0
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Certainly! Here is a Calculus question
along with the step-by-step solution in LateX code:
Question 13: Find the derivative of the function f(x)=3x4−2x3+
5x2−7x+ 9.
Step-by-step solution:
Given function: f(x)=3x4−2x3+ 5x2−7x+ 9
To find the derivative of f(x), we differentiate each term with re-
spect to x.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
= 12x3−6x2+ 10x−7+0
= 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
12
Question 14
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+9
To find the derivative f′(x), we differentiate each term of the func-
tion with respect to xusing the power rule of differentiation.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3) = 3 · −2x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1· −7x1−1=−7
5. Differentiate the constant term 9: Since the derivative of a
constant is 0, the derivative of the constant term 9is 0.
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 14: Find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9.
Step-by-step solution: Given function: f(x)=3x4−2x3+5x2−7x+9
To find the derivative f′(x), we differentiate each term of the func-
tion with respect to xusing the power rule of differentiation.
1. Differentiate 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Differentiate −2x3:
d
dx (−2x3) = 3 · −2x3−1=−6x2
3. Differentiate 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Differentiate −7x:
d
dx (−7x)=1· −7x1−1=−7
13
5. Differentiate the constant term 9: Since the derivative of a
constant is 0, the derivative of the constant term 9is 0.
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 15
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 9, we will apply the power rule of differentiation:
1. Calculate the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Calculate the derivative of the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Calculate the derivative of the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
4. Calculate the derivative of the term −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term 9is zero.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 15: Find the derivative of the function f(x) = 3x4−2x3+
5x2−7x+ 9.
Step-by-step solution: To find the derivative of the function f(x) =
3x4−2x3+ 5x2−7x+ 9, we will apply the power rule of differentiation:
1. Calculate the derivative of the term 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Calculate the derivative of the term −2x3:
d
dx (−2x3)=3·(−2)x3−1=−6x2
3. Calculate the derivative of the term 5x2:
d
dx (5x2)=2·5x2−1= 10x
14
4. Calculate the derivative of the term −7x:
d
dx (−7x)=1·(−7)x1−1=−7
5. The derivative of a constant term 9is zero.
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is:
f′(x) = 12x3−6x2+ 10x−7
Question 16
Step-by-step solution:
f(x) = 3x4−2x3+ 5x2−7x+ 9
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.Question 16: Find the derivative of the
function f(x) = 3x4−2x3+ 5x2−7x+ 9.
Step-by-step solution:
f(x) = 3x4−2x3+ 5x2−7x+ 9
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+9
is f′(x) = 12x3−6x2+ 10x−7.
Question 17
Question 17: 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 will differentiate each term with respect to
xusing the power rule for derivatives:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
15
Now, applying the power rule:
f′(x)=3·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
Simplifying the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Sure! Here is a question on Calculus:
Question 17: 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 will differentiate each term with respect to
xusing the power rule for derivatives:
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (9)
Now, applying the power rule:
f′(x)=3·4x4−1−2·3x3−1+ 5 ·2x2−1−7·1x1−1+ 0
Simplifying the expression:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function is:
f′(x) = 12x3−6x2+ 10x−7
Question 18
Question 18: Find the derivative of the function f(x)=3x2+ 5x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 5x−2,
we will apply the power rule for derivatives.
Step 1: Find the derivative of 3x2.
f′(x) = 3 ·2x2−1= 6x
Step 2: Find the derivative of 5x.
f′(x) = 5 ·1x1−1= 5
16
Step 3: Find the derivative of the constant term −2.
f′(x)=0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Feel free to reach out if you need further assistance or more ques-
tions!Sure! Here is a Calculus question along with its step-by-step
solution in LateX code:
Question 18: Find the derivative of the function f(x)=3x2+ 5x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 5x−2,
we will apply the power rule for derivatives.
Step 1: Find the derivative of 3x2.
f′(x)=3·2x2−1= 6x
Step 2: Find the derivative of 5x.
f′(x) = 5 ·1x1−1= 5
Step 3: Find the derivative of the constant term −2.
f′(x)=0
Therefore, the derivative of the function f(x)=3x2+ 5x−2is:
f′(x)=6x+ 5
Feel free to reach out if you need further assistance or more ques-
tions!
Question 19
Question 19: Find the derivative of the function f(x)=3x2+ 4x−2.
Solution: To find the derivative of the function f(x) = 3x2+ 4x−2,
we can use the power rule for differentiation.
Step 1: Find the derivative of the term 3x2.
d
dx (3x2)=2×3×x2−1= 6x
Step 2: Find the derivative of the term 4x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −2.
d
dx (−2) = 0
17
Step 4: Combine the derivatives of individual terms to find the
derivative of the function f(x) = 3x2+ 4x−2.
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.
This completes the solution.Sure! Here is a Calculus question
along with its step-by-step solution in LateX code:
Question 19: Find the derivative of the function f(x)=3x2+ 4x−2.
Solution: To find the derivative of the function f(x)=3x2+ 4x−2,
we can use the power rule for differentiation.
Step 1: Find the derivative of the term 3x2.
d
dx (3x2)=2×3×x2−1= 6x
Step 2: Find the derivative of the term 4x.
d
dx (4x)=4
Step 3: Find the derivative of the constant term −2.
d
dx (−2) = 0
Step 4: Combine the derivatives of individual terms to find the
derivative of the function f(x) = 3x2+ 4x−2.
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.
This completes the solution.
Question 20
f(x)=3x4−8x3+ 2x2−5x+ 1
Step-by-step solution: Let’s differentiate using the power rule for
derivatives, which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−8x3+ 2x2−5x+ 1
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
18
2. Find the derivative of −8x3:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Find the derivative of 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. Find the derivative of −5x:
d
dx (−5x)=1·(−5)x1−1=−5
5. The derivative of a constant is 0:
d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−8x3+2x2−5x+1
is:
f′(x) = 12x3−24x2+ 4x−5
Question 20: Calculate the derivative of the following function:
f(x)=3x4−8x3+ 2x2−5x+ 1
Step-by-step solution: Let’s differentiate using the power rule for
derivatives, which states that the derivative of xnis nxn−1.
Given function: f(x)=3x4−8x3+ 2x2−5x+ 1
1. Find the derivative of 3x4:
d
dx (3x4) = 4 ·3x4−1= 12x3
2. Find the derivative of −8x3:
d
dx (−8x3) = 3 ·(−8)x3−1=−24x2
3. Find the derivative of 2x2:
d
dx (2x2)=2·2x2−1= 4x
4. Find the derivative of −5x:
d
dx (−5x)=1·(−5)x1−1=−5
5. The derivative of a constant is 0:
d
dx (1) = 0
Therefore, the derivative of the function f(x)=3x4−8x3+2x2−5x+1
is:
f′(x) = 12x3−24x2+ 4x−5
19
Question 21
Question 21: Find the derivative of the function f(x)=2x3−5x2+
4x−7.
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule of
differentiation.
f′(x) = d
dx (2x3)−d
dx (5x2) + d
dx (4x)−d
dx (7)
= 2 ·3x3−1−5·2x2−1+ 4 ·1x1−1−0
= 6x2−10x+ 4
Therefore, the derivative of the function f(x)=2x3−5x2+ 4x−7is
f′(x) = 6x2−10x+ 4.Certainly! Here is a Calculus question along with
its step-by-step solution in LateX code:
Question 21: Find the derivative of the function f(x)=2x3−5x2+
4x−7.
Step-by-step solution: To find the derivative of the function f(x),
we will differentiate each term separately using the power rule of
differentiation.
f′(x) = d
dx (2x3)−d
dx (5x2) + d
dx (4x)−d
dx (7)
= 2 ·3x3−1−5·2x2−1+ 4 ·1x1−1−0
= 6x2−10x+ 4
Therefore, the derivative of the function f(x)=2x3−5x2+ 4x−7is
f′(x) = 6x2−10x+ 4.
Question 22
Question 22:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 4
Step-by-step solution:
To find the derivative of the function f(x) = 3x4−2x3+ 5x2−7x+ 4,
we need to differentiate each term separately using the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (4)
Now, let’s calculate the derivatives of each term:
f′(x) = 12x3−6x2+ 10x−7
20
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is f′(x) = 12x3−6x2+ 10x−7.
This is the solution to question 22.Certainly! Here is a Calculus
question along with its step-by-step solution in LateX code:
Question 22:
Find the derivative of the following function:
f(x)=3x4−2x3+ 5x2−7x+ 4
Step-by-step solution:
To find the derivative of the function f(x)=3x4−2x3+ 5x2−7x+ 4,
we need to differentiate each term separately using the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7x) + d
dx (4)
Now, let’s calculate the derivatives of each term:
f′(x) = 12x3−6x2+ 10x−7
Therefore, the derivative of the function f(x)=3x4−2x3+5x2−7x+4
is f′(x) = 12x3−6x2+ 10x−7.
This is the solution to question 22.
Question 23
Question 23: Find the derivative of the function f(x)=3x4−2x3+
5x2−7.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we will apply the power rule for differentiation.
Step 1: Calculate the derivative by applying the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7)
Step 2: Apply the power rule to find the derivatives.
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−0
f′(x) = 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.Sure! Here’s a Calculus question along with
the step-by-step solution in LateX code:
Question 23: Find the derivative of the function f(x)=3x4−2x3+
5x2−7.
Solution: To find the derivative of the function f(x) = 3x4−2x3+
5x2−7, we will apply the power rule for differentiation.
21
Step 1: Calculate the derivative by applying the power rule.
f′(x) = d
dx (3x4)−d
dx (2x3) + d
dx (5x2)−d
dx (7)
Step 2: Apply the power rule to find the derivatives.
f′(x) = 3 ·4x4−1−2·3x3−1+ 5 ·2x2−1−0
f′(x) = 12x3−6x2+ 10x
Therefore, the derivative of the function f(x) = 3x4−2x3+ 5x2−7
is f′(x) = 12x3−6x2+ 10x.
Question 24
Solution:
To find the derivative of the function f(x)=3x2−4x+ 5, we will
use the power rule for differentiation.
Given function, f(x)=3x2−4x+ 5
Now, differentiate each term of the function with respect to xusing
the power rule:
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=2·3x2−1−4·1x1−1+ 0
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.24. Find the derivative of the function f(x)=3x2−4x+5.
Solution:
To find the derivative of the function f(x)=3x2−4x+ 5, we will
use the power rule for differentiation.
Given function, f(x)=3x2−4x+ 5
Now, differentiate each term of the function with respect to xusing
the power rule:
f′(x) = d
dx (3x2)−d
dx (4x) + d
dx (5)
f′(x)=2·3x2−1−4·1x1−1+ 0
f′(x)=6x−4
Therefore, the derivative of the function f(x) = 3x2−4x+ 5 is
f′(x) = 6x−4.
22
Question 25
Find the derivative of the function f(x) = 5x3−2x2+ 4x−7.
Step-by-step solution:
To find the derivative of the function f(x) = 5x3−2x2+ 4x−7, we
need to differentiate each term with respect to x.
1. Differentiate 5x3with respect to x:
d
dx (5x3) = 3 ·5x3−1= 15x2
2. Differentiate −2x2with respect to x:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate 4xwith respect to x:
d
dx (4x)=4
4. Differentiate the constant term −7with respect to x: Since −7
is a constant, its derivative is 0.
Therefore, the derivative of the function f(x)=5x3−2x2+ 4x−7is:
f′(x) = 15x2−4x+ 4
So, the derivative of the function is f′(x) = 15x2−4x+ 4.Question
25:
Find the derivative of the function f(x) = 5x3−2x2+ 4x−7.
Step-by-step solution:
To find the derivative of the function f(x) = 5x3−2x2+ 4x−7, we
need to differentiate each term with respect to x.
1. Differentiate 5x3with respect to x:
d
dx (5x3) = 3 ·5x3−1= 15x2
2. Differentiate −2x2with respect to x:
d
dx (−2x2)=2·(−2)x2−1=−4x
3. Differentiate 4xwith respect to x:
d
dx (4x)=4
4. Differentiate the constant term −7with respect to x: Since −7
is a constant, its derivative is 0.
Therefore, the derivative of the function f(x)=5x3−2x2+ 4x−7is:
f′(x) = 15x2−4x+ 4
So, the derivative of the function is f′(x) = 15x2−4x+ 4.
23
Question 26
Question 26: Find the derivative of the following function: f(x) =
3x4+ 2x3−5x2+ 8.
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+8, we will differentiate each term separately using the power rule
for derivatives.
Given function: f(x)=3x4+ 2x3−5x2+ 8.
The derivative of 3x4is obtained by multiplying the coefficient by
the power and then reducing the power by 1.
d
dx (3x4)=4·3x4−1= 12x3.
The derivative of 2x3is calculated in a similar manner using the
power rule.
d
dx (2x3)=3·2x3−1= 6x2.
Similarly, the derivative of −5x2is found as follows:
d
dx (−5x2)=2·(−5)x2−1=−10x.
The derivative of a constant term like 8 is always 0.
d
dx (8) = 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 8
is:
f′(x) = 12x3+ 6x2−10x.
Thus, the derivative of the function is f′(x) = 12x3+6x2−10x.Certainly!
Here’s a Calculus question along with its step-by-step solution in La-
teX code:
Question 26: Find the derivative of the following function: f(x) =
3x4+ 2x3−5x2+ 8.
Solution: To find the derivative of the function f(x) = 3x4+ 2x3−
5x2+8, we will differentiate each term separately using the power rule
for derivatives.
Given function: f(x)=3x4+ 2x3−5x2+ 8.
The derivative of 3x4is obtained by multiplying the coefficient by
the power and then reducing the power by 1.
d
dx (3x4)=4·3x4−1= 12x3.
The derivative of 2x3is calculated in a similar manner using the
power rule.
d
dx (2x3)=3·2x3−1= 6x2.
Similarly, the derivative of −5x2is found as follows:
d
dx (−5x2)=2·(−5)x2−1=−10x.
The derivative of a constant term like 8 is always 0.
d
dx (8) = 0.
Therefore, the derivative of the function f(x) = 3x4+ 2x3−5x2+ 8
is:
f′(x) = 12x3+ 6x2−10x.
Thus, the derivative of the function is f′(x) = 12x3+ 6x2−10x.
24
Question 27
Question 27: Find the derivative of the function f(x)=3x2+ 5x+ 8.
Solution: To find the derivative of the function f(x) = 3x2+ 5x+ 8,
we use the power rule of differentiation.
Step 1: Identify the function The function given is f(x)=3x2+
5x+ 8.
Step 2: Apply the power rule The power rule states that if f(x) =
axn, then f′(x) = n·axn−1. Using this rule, we differentiate each term
of the function separately.
Step 3: Differentiate each term - Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
- Differentiate the term 5x:
d
dx (5x)=1·5x1−1= 5
- Differentiate the constant term 8: The derivative of a constant is 0.
Step 4: Combine the derivatives Putting together the derivatives
of each term, we get the derivative of the function:
f′(x)=6x+ 5
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 8 is
f′(x) = 6x+ 5.Sure! Here is a Calculus question along with its step-
by-step solution in LateX code:
Question 27: Find the derivative of the function f(x)=3x2+ 5x+ 8.
Solution: To find the derivative of the function f(x) = 3x2+ 5x+ 8,
we use the power rule of differentiation.
Step 1: Identify the function The function given is f(x)=3x2+
5x+ 8.
Step 2: Apply the power rule The power rule states that if f(x) =
axn, then f′(x) = n·axn−1. Using this rule, we differentiate each term
of the function separately.
Step 3: Differentiate each term - Differentiate the term 3x2:
d
dx (3x2)=2·3x2−1= 6x
- Differentiate the term 5x:
d
dx (5x)=1·5x1−1= 5
- Differentiate the constant term 8: The derivative of a constant is 0.
Step 4: Combine the derivatives Putting together the derivatives
of each term, we get the derivative of the function:
f′(x)=6x+ 5
25
Therefore, the derivative of the function f(x) = 3x2+ 5x+ 8 is
f′(x) = 6x+ 5.
Question 28
Question 28: Find the derivative of the function f(x)=3x2+ 4x−7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 4x−7, we need to differentiate each term with respect to xusing
the power rule.
Given function: f(x)=3x2+ 4x−7
Step 1: Differentiate the term 3x2:
f′(x) = d
dx (3x2)
f′(x)=2·3x2−1
f′(x)=6x
Step 2: Differentiate the term 4x:
f′(x) = d
dx (4x)
f′(x) = 4 ·1
f′(x)=4
Step 3: Differentiate the constant term −7: The derivative of a
constant is zero, so it does not contribute to the derivative of the
function.
Step 4: Combine the derivatives of each term to find the derivative
of the function f(x) = 3x2+ 4x−7:
f′(x)=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−7is
f′(x)=6x+ 4.Certainly! Here’s a Calculus question along with step-
by-step solution in LateX code:
Question 28: Find the derivative of the function f(x)=3x2+ 4x−7.
Step-by-step Solution: To find the derivative of the function f(x) =
3x2+ 4x−7, we need to differentiate each term with respect to xusing
the power rule.
Given function: f(x)=3x2+ 4x−7
Step 1: Differentiate the term 3x2:
f′(x) = d
dx (3x2)
f′(x)=2·3x2−1
26
f′(x)=6x
Step 2: Differentiate the term 4x:
f′(x) = d
dx (4x)
f′(x) = 4 ·1
f′(x)=4
Step 3: Differentiate the constant term −7: The derivative of a
constant is zero, so it does not contribute to the derivative of the
function.
Step 4: Combine the derivatives of each term to find the derivative
of the function f(x) = 3x2+ 4x−7:
f′(x)=6x+ 4
Therefore, the derivative of the function f(x) = 3x2+ 4x−7is
f′(x) = 6x+ 4.
Question 29
Question 29: Find the derivative of the following function: f(x) =
3x2+ 4x−5
Solution:
To find the derivative of f(x)=3x2+4x−5, we will apply the power
rule for derivatives and the constant multiple rule.
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4x:
d
dx (4x)=4
Step 3: Combine the derivatives of individual terms:
f′(x)=6x+ 4
Therefore, the derivative of f(x)=3x2+ 4x−5is f′(x)=6x+
4.Certainly! Here is a calculus question along with its step-by-step
solution written in LateX code:
Question 29: Find the derivative of the following function: f(x) =
3x2+ 4x−5
Solution:
To find the derivative of f(x)=3x2+4x−5, we will apply the power
rule for derivatives and the constant multiple rule.
27
Step 1: Find the derivative of 3x2:
d
dx (3x2)=2·3x2−1= 6x
Step 2: Find the derivative of 4x:
d
dx (4x)=4
Step 3: Combine the derivatives of individual terms:
f′(x)=6x+ 4
Therefore, the derivative of f(x)=3x2+ 4x−5is f′(x)=6x+ 4.
Question 30
Question 30: Find the derivative of the function f(x)=3x2+ 4x−5
using the power rule.
Solution: To find the derivative of the function using the power
rule, we differentiate each term separately. The power rule states
that if f(x) = axn, then f′(x) = anxn−1.
Given function: f(x)=3x2+ 4x−5
Differentiate each term:
d
dx (3x2)=2·3x2−1= 6x
d
dx (4x)=1·4x1−1= 4
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
This is the final answer.
f′(x)=6x+ 4
Certainly! Here is a calculus question along with the step-by-step
solution in LateX code:
Question 30: Find the derivative of the function f(x)=3x2+ 4x−5
using the power rule.
Solution: To find the derivative of the function using the power
rule, we differentiate each term separately. The power rule states
that if f(x) = axn, then f′(x) = anxn−1.
28
Given function: f(x)=3x2+ 4x−5
Differentiate each term:
d
dx (3x2)=2·3x2−1= 6x
d
dx (4x)=1·4x1−1= 4
d
dx (−5) = 0
Therefore, the derivative of the function f(x)=3x2+ 4x−5is:
f′(x)=6x+ 4
This is the final answer.
f′(x)=6x+ 4
29
