Master the Chain Rule: Enhance Your Calculus I Skills

Question 1

State the formula for the Chain Rule.

Accepted Answer

d(f(x))/dx = f'(x) * g'(x)

Answer

d(f(x))/dx = f'(x) * g(x)

Answer

d(f(x))/dx = f(x) * g'(x)

Answer

d(f(x))/dx = g'(x) * f(x)

Question 2

Use the Chain Rule to find the derivative of f(x) = (x^2 + 1)^3.

Accepted Answer

f'(x) = 6x(x^2 + 1)^2

Answer

f'(x) = 3x^2(x^2 + 1)

Answer

f'(x) = 3(x^2 + 1)^2

Answer

f'(x) = 6x(x^2 + 2)

Question 3

Determine the derivative of h(x) = sin(2x + 1).

Accepted Answer

h'(x) = 2cos(2x + 1)

Answer

h'(x) = cos(2x + 1)

Answer

h'(x) = 2sin(2x + 1)

Answer

h'(x) = cos(2x)

Question 4

Identify the function that cannot be differentiated using the Chain Rule.

Accepted Answer

h(x) = |x|

Answer

f(x) = x^3 + e^x

Answer

g(x) = ln(x^2 + 1)

Answer

j(x) = sqrt(x^2 - 4)

Question 5

Determine the derivative of g(x) = arctan(3x - 2).

Accepted Answer

g'(x) = 3/(1 + (3x - 2)^2)

Answer

g'(x) = 3/(3x - 2)

Answer

g'(x) = 1/(1 + (3x - 2)^2)

Answer

g'(x) = (3x - 2)/(1 + (3x - 2)^2)

Question 6

Determine the partial derivative of f(x, y) = x^2y with respect to y.

Accepted Answer

f_y(x, y) = 2xy

Answer

f_x(x, y) = 2xy

Answer

f_x(x, y) = x^2

Answer

f_y(x, y) = x^2

Question 7

Can the Chain Rule be applied to find the derivative of f(x) = 1 + |x| at x = 0?

Accepted Answer

No

Answer

Yes

Question 8

Identify the most challenging aspect of applying the Chain Rule?

Accepted Answer

Identifying the inner and outer functions

Answer

Applying the derivative of the outer function

Answer

Multiplying the derivatives

Answer

Simplifying the result

Question 9

Consider the function h(x) = sqrt(x^2 + 1). Using the Chain Rule, find h'(x).

Accepted Answer

(1/2)(x^2 + 1)^(-1/2) * x

Answer

(x^2 + 1)^(-1/2)

Answer

x

Answer

(1/2)(x^2 + 1)^(-1/2) * 2x

Question 10

Find the derivative of h(x) = e^(sin(x)) using the Chain Rule.

Accepted Answer

e^(sin(x)) * cos(x)

Answer

sin(x) * e^(sin(x))

Answer

cos(x) * e^(sin(x))

Answer

e^(sin(x))

Question 11

Let f(x) = x^2 and g(x) = sqrt(x). Use the Chain Rule to determine the derivative of the composition f(g(x)).

Accepted Answer

1 / (2 * sqrt(x))

Answer

2 * sqrt(x)

Answer

1 / (2 * x)

Answer

2 * x

Question 12

Find the derivative of h(x) = arctan(x^2 + 1) using the Chain Rule.

Accepted Answer

2x / (1 + x^4)

Answer

1 / (x^2 + 1)

Answer

1 / (2 * x * (1 + x^4))

Answer

2x / (1 + x^2)

Question 13

Let f(x) = cos(x) and g(x) = x^3. Find the derivative of the composition g(f(x)) using the Chain Rule.

Accepted Answer

-sin(x) * 3x^2

Answer

-sin(x^3)

Answer

-cos(x) * 3x^2

Answer

sin(x^3) * 3x^2

Question 14

Let f(x) = x^2 and g(x) = sin(x). Find the derivative of the composition f(g(x)) using the Chain Rule.

Accepted Answer

cos(x^2) * 2x

Answer

cos(x) * 2x

Answer

sin(x^2)

Answer

2x * sin(x)

Question 15

If f(x) = sin(x^3), what is f'(x)?

Accepted Answer

3x^2 * cos(x^3)

Answer

cos(x^3)

Answer

sin(x^3) * 3x^2

Answer

3cos(x^3)

Question 16

What is the derivative of f(x) = e^(3x - 1)?

Accepted Answer

3e^(3x - 1)

Answer

e^(3x - 1)

Answer

3x * e^(3x - 1)

Answer

e^(3x - 1) / 3

Question 17

Find the derivative of f(x) = (tan(x))^5.

Accepted Answer

5(tan(x))^4 * sec^2(x)

Answer

(tan(x))^4 * sec^2(x)

Answer

5(tan(x))^4 * sec(x)

Answer

(tan(x))^5

Question 18

What is the derivative of f(x) = sqrt(sin(x^2 + 3))?

Accepted Answer

(1/2) * (cos(x^2 + 3) / sqrt(sin(x^2 + 3))) * 2x

Answer

(1/2) * (sin(x^2 + 3) / sqrt(cos(x^2 + 3))) * 2x

Answer

(cos(x^2 + 3) / sqrt(sin(x^2 + 3))) * 2x

Answer

(sin(x^2 + 3) / sqrt(cos(x^2 + 3))) * 2x

Question 19

What is the derivative of f(x) = 10^(x - 1)?

Accepted Answer

10 * ln(10) * (x - 1)

Answer

10^(x - 1)

Answer

10 * ln(10)

Answer

ln(10) * (x - 1)

Question 20

Find the derivative of f(x) = e^(x^2 + 1).

Accepted Answer

2xe^(x^2 + 1)

Answer

(x^2 + 1)e^(x^2 + 1)

Answer

e^(x^2 + 1)

Answer

xe^(x^2 + 1)

Question 21

Which of the following steps is NOT involved in applying the Chain Rule?

Accepted Answer

Simplifying the derivative

Answer

Differentiating the outer function with respect to the inner function

Answer

Multiplying the derivatives of the inner and outer functions

Answer

Identifying the inner and outer functions

Question 22

Find the derivative of f(x) = 3x^2 * sin(x).

Accepted Answer

6xsin(x) + 3x^2cos(x)

Answer

6xcos(x) - 3x^2sin(x)

Answer

3x^2cos(x) + sin(x)

Answer

cos(x) + sin(x)

Question 23

When should the Chain Rule be applied?

Accepted Answer

To find the derivative of a composition of functions

Answer

To find the derivative of a polynomial

Answer

To find the derivative of an exponential function

Answer

To find the derivative of a trigonometric function

Question 24

Find the derivative of f(x) = sin(x^2 + 1) using the Chain Rule.

Accepted Answer

2x * cos(x^2 + 1)

Answer

cos(x^2 + 1)

Answer

2cos(x)

Question 25

Define the Chain Rule and explain its importance in finding the derivative of a composite function.

Accepted Answer

The Chain Rule is a mathematical technique used to find the derivative of a function composed of two or more other functions.

Answer

The Chain Rule can only be used to differentiate continuous functions.

Answer

The Chain Rule is a special case of the product rule.

Answer

The Chain Rule cannot be used to find the derivative of an inverse function.

Question 26

Find the derivative of f(x) = (x^2 + 1)^5 using the Chain Rule.

Accepted Answer

10x(x^2 + 1)^4

Answer

5x(x^2 + 1)^3

Answer

5(2x)

Answer

10(x^2 + 1)^3

Question 27

Calculate the derivative of g(t) = sin(t^3 - 1) using the Chain Rule.

Accepted Answer

(3t^2)cos(t^3 - 1)

Answer

3tcos(t^3 - 1)

Answer

cos(t^3 - 1)

Answer

(t^3 - 1)cos(t^3 - 1)

Question 28

Find the derivative of f(x) = ln(x^2 - 4) using the Chain Rule.

Accepted Answer

(2x)/(x^2 - 4)

Answer

1/(x^2 - 4)

Answer

1/(2x)

Answer

2x

Question 29

Calculate the derivative of g(x) = arctan(3x^2 + 2) using the Chain Rule.

Accepted Answer

6x/(3x^2 + 2)^2

Answer

1/(3x^2 + 2)

Answer

3/(3x^2 + 2)^2

Answer

6x/(3x^2 + 2)

Question 30

Find the derivative of h(x) = (x^2 + 1)^3 * sin(x) using the Chain Rule.

Accepted Answer

3(x^2 + 1)^2 * 2x * sin(x) + (x^2 + 1)^3 * cos(x)

Answer

3(x^2 + 1) * 2x * sin(x) + (x^2 + 1)^3 * cos(x)

Answer

3(x^2 + 1)^2 * 2x * cos(x) + (x^2 + 1)^3 * sin(x)

Answer

(x^2 + 1)^3 * cos(x)

Question 31

Determine the derivative of f(x) = sqrt(x^3 + 2x - 1) using the Chain Rule.

Accepted Answer

(3x^2 + 2)/(2 * sqrt(x^3 + 2x - 1))

Answer

1/(2 * sqrt(x^3 + 2x - 1))

Answer

(3x^2 + 2)

Answer

1/(3x^2 + 2 * sqrt(x^3 + 2x - 1))

Question 32

Find the derivative of g(x) = (1 + e^x)^2 * ln(x + 1) using the Chain Rule.

Accepted Answer

(1 + e^x)^2 * 1/(x + 1) + 2(1 + e^x) * e^x * ln(x + 1)

Answer

(1 + e^x)^2

Answer

(1 + e^x) * 1/(x + 1)

Question 33

Determine the derivative of f(x) = (x^2 + 1)/(sqrt(x) + 1) using the Chain Rule.

Accepted Answer

((2x)(sqrt(x) + 1) - (x^2 + 1)(1/2x^(1/2)))/(sqrt(x) + 1)^2

Answer

(x^2 + 1)/(sqrt(x) + 1)

Answer

(2x)(sqrt(x) + 1) - (x^2 + 1)(1/2x^(1/2))

Question 34

Which of the following correctly represents the formula for the Chain Rule?

Accepted Answer

dy/dx = (dy/du) * (du/dx)

Answer

dy/dx = (du/dy) * (du/dx)

Answer

dy/dx = (dy/du) * (dx/du)

Answer

dy/dx = (dy/du)^2 * (du/dx)

Question 35

Use the Chain Rule to find the derivative of f(x) = cos(e^(3x)).

Accepted Answer

-3e^(3x) * sin(e^(3x))

Answer

e^(3x) * cos(e^(3x))

Answer

3e^(3x) * cos(e^(3x))

Answer

-sin(e^(3x))

Question 36

Which formula represents the Chain Rule for the nth derivative?

Accepted Answer

(d^n/dx^n)f(g(x)) = f^(n)(g(x)) * (d^n/dx^n)g(x)

Answer

(d^n/dx^n)f(g(x)) = f^(n)(g(x)) + (d^n/dx^n)g(x)

Answer

(d^n/dx^n)f(g(x)) = f(g(x))^(n+1)

Answer

(d^n/dx^n)f(g(x)) = f^(n-1)(g(x)) * (d^n/dx^n)g(x)

Question 37

Apply the Chain Rule to find the derivative of f(x) = ln(x^2 + 2x + 1).

Accepted Answer

(2x + 2)/(x^2 + 2x + 1)

Answer

2ln(x^2 + 2x + 1)

Answer

1/(x^2 + 2x + 1)

Answer

(x^2 + 2x + 1)

Question 38

What is a limitation of the Chain Rule?

Accepted Answer

It cannot be applied to functions that are not differentiable

Answer

It is not applicable to trigonometric functions

Answer

It can only be applied to polynomials

Question 39

Use the Chain Rule to find the derivative of f(x) = e^(cos(2x^2 + 3x)).

Accepted Answer

(-4x * sin(2x^2 + 3x) + 3 * cos(2x^2 + 3x)) * e^(cos(2x^2 + 3x))

Answer

(-4x * cos(2x^2 + 3x) + 3 * sin(2x^2 + 3x)) * e^(cos(2x^2 + 3x))

Answer

e^(cos(2x^2 + 3x)) * (-4x * sin(2x^2 + 3x) + 3 * cos(2x^2 + 3x))

Answer

2e^(cos(2x^2 + 3x)) * (-4x * sin(2x^2 + 3x) + 3 * cos(2x^2 + 3x))

Question 40

If f(x) is a differentiable function and g(x) = f(2x + 3), find g'(x).

Accepted Answer

2f'(2x + 3)

Answer

f'(2x)

Answer

f'(2x + 3)

Answer

2f'(x)

Question 41

Calculate the derivative of g(x) = sin(x^2 + 1) using the Chain Rule.

Accepted Answer

2x * cos(x^2 + 1)

Answer

2 * cos(x^2 + 1)

Answer

cos(x^2 + 1)

Answer

x * cos(x^2 + 1)

Question 42

Find the derivative of the function g(x) = arctan(cos(2x)) using the Chain Rule.

Accepted Answer

-2sin(2x)/(1 + cos^2(2x))

Answer

2sin(2x)/(1 + cos^2(2x))

Answer

-sin(2x)/(1 + cos^2(2x))

Answer

sin(2x)/(1 + cos^2(2x))

Question 43

Find the derivative of h(x) = (e^x + ln(x))^2 using the Chain Rule.

Accepted Answer

2(e^x + ln(x)) * (e^x + 1/x)

Answer

(e^x + ln(x)) * (e^x - 1/x)

Answer

(e^x + ln(x)) * (e^x + 1/x)

Answer

2(e^x + ln(x)) * (e^x - 1/x)

Question 44

Determine the derivative of the function f(x) = (x^2 + 5x)^3.

Accepted Answer

3(x^2 + 5x)^2 (2x + 5)

Answer

3x(x^2 + 5x)

Answer

(x^2 + 5x)^2

Answer

None of these

Question 45

Calculate the derivative of the function f(x) = ln(sin(x^2)).

Accepted Answer

2xcot(x^2) / sin(x^2)

Answer

cos(x^2) / sin(x^2)

Answer

ln(sin(x^2))

Question 46

Find the derivative of the function g(x) = (e^x + sin(x))^2.

Accepted Answer

(e^x + sin(x)) * (e^x + cos(x))

Answer

None of these

Answer

2(e^x + sin(x))

Answer

(e^x + sin(x)) * (e^x - cos(x))

Question 47

Evaluate the derivative of the function f(x) = tan(√(x^2 + 1)).

Accepted Answer

(x/(√(x^2 + 1))) * sec^2(√(x^2 + 1))

Answer

None of these

Answer

sec^2(√(x^2 + 1))

Answer

(√(x^2 + 1))/(x) * cot^2(√(x^2 + 1))

Question 48

Calculate the derivative of the function y = (cos(x^3) + x^2)^4.

Accepted Answer

4(cos(x^3) + x^2)^3 (-3x^2sin(x^3) + x)

Answer

(cos(x^3) + x^2)^3

Answer

4(cos(x^3) + x^2)^3

Question 49

Find the derivative of the function f(x) = √(e^(x^2) + ln(x)).

Accepted Answer

((xe^(x^2)) / √(e^(x^2) + ln(x))) + ((1/x) / √(e^(x^2) + ln(x)))

Answer

1 / √(e^(x^2) + ln(x))

Answer

e^(x^2) / √(e^(x^2) + ln(x))

Question 50

Let f(x) = x^2 + 3 and g(x) = sin(x). Find the derivative of the composite function h(x) = f(g(x)).

Accepted Answer

(cos(x)) * 2x

Answer

2x * sin(x)

Answer

cos(x) * (x^2 + 3)

Answer

sin(x) * (x^2 + 3)

Question 51

Let f(x) = e^x and g(x) = ln(x). Find the derivative of the composite function h(x) = g(f(x)).

Accepted Answer

1/x

Answer

x

Answer

e^x

Answer

ln(x)

Question 52

Find the derivative of the function h(x) = (x^2 - 1)^(1/3).

Accepted Answer

(2x) / (3(x^2 - 1)^(2/3))

Answer

(x^2 - 1) / (3x)

Answer

(3x) / (x^2 - 1)

Answer

(2x) / (3(x^2 - 1))

Question 53

Let f(x) = tan(x) and g(x) = cos(x). Find the derivative of the composite function h(x) = f(g(x)).

Accepted Answer

-sin(x)*sin(cos(x))

Answer

-cos(x)*cos(sin(x))

Answer

cos(x)*cos(sin(x))

Answer

sin(x)*sin(cos(x))

Question 54

Let f(x) = sqrt(x) and g(x) = x^3 - 1. Find the derivative of the composite function h(x) = f(g(x)).

Accepted Answer

(3x^2) / (2sqrt(x^3 - 1))

Answer

(sqrt(x^3 - 1)) / (2x)

Answer

(x^3 - 1) / (2sqrt(x))

Answer

(2x) / (sqrt(x^3 - 1))

Question 55

Which of the following is a step in applying the Chain Rule?

Accepted Answer

Find the derivative of the inner function.

Answer

Substitute the expression of the inner function into the derivative of the outer function.

Answer

Multiply the derivative of the inner function by the derivative of the outer function.

Question 56

Find the derivative of the function h(x) = |x + 1|.

Accepted Answer

1 if x >= -1, -1 if x < -1

Answer

-1 if x > 0, 1 if x < 0

Answer

x if x >= -1, -x if x < -1

Answer

1 if x > 0, -1 if x < 0

Question 57

Let f(x) = sin(x) and g(x) = x^2. Find the derivative of the composite function h(x) = f(g(x)).

Accepted Answer

2x*cos(x^2)

Answer

cos(x^2)/2x

Answer

sin(x^2)/x

Answer

-2x*cos(x^2)

Question 58

Which of the following functions requires the application of the Chain Rule to find its derivative?

Accepted Answer

y = e^(x^2)

Answer

y = x^3 + 2x

Answer

y = sin(x) + cos(x)

Answer

y = |x|

Question 59

Find the derivative of the function h(x) = arctan(x^2).

Accepted Answer

2x/(1 + x^4)

Answer

2x/(1 - x^4)

Answer

1/(1 + x^4)

Answer

1/(1 - x^4)