Master the Chain Rule: Enhance Your Calculus III Skills

Question 1

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

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

sin(2x)

Answer

x^2 cos(x^2)

Question 2

Find dy/dx for the function y = cos(3x - 2).

Accepted Answer

3sin(3x - 2)

Answer

-3sin(3x - 2)

Answer

-sin(3x - 2)

Answer

sin(3x - 2)

Question 3

Find the derivative of the function g(x) = tan(arcsin(x)).

Accepted Answer

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

Answer

1/(1 - x^2)

Answer

(1 + x^2)/2x

Answer

1/(1 + x^2)

Question 4

Find the equation of the tangent line to the curve y = x^3 - 2x^2 + 1 at the point (1, 0).

Accepted Answer

y = 3x - 2

Answer

y = -x + 2

Answer

y = 2x + 1

Answer

y = x - 2

Question 5

Which of the following is NOT a step in applying the Chain Rule?

Accepted Answer

Add the derivatives of both functions.

Answer

Identify the outer and inner functions.

Answer

Multiply the result by the derivative of the inner function.

Answer

Apply the derivative of the outer function to the inner function.

Question 6

Calculate the derivative of f(x) = ln((x^3 + 1)^2).

Accepted Answer

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

Answer

2x/(x^3 + 1)

Answer

(x^3 + 1)^2

Answer

x^3

Question 7

Let f(x) = sin(x^2). Find f'(x) using the Chain Rule.

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

x^2 cos(x^2)

Answer

2x^2 cos(x^2)

Question 8

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

Accepted Answer

2xe^(x^2 - 1)

Answer

xe^(x^2 - 1)

Answer

e^(x^2 - 1)

Answer

2e^(x^2 - 1)

Question 9

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

Accepted Answer

(B) (6x^2 cos(x^2))

Answer

(A) (2x cos(x^2))

Answer

(C) (sin(x^2))

Answer

(D) (3x^2 sin(x^2))

Question 10

Calculate the derivative of f(x) = sin(x^2 + 3x).

Accepted Answer

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

Answer

cos(x^2 + 3x)

Answer

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

Answer

2xsin(x^2 + 3x)

Question 11

Determine dy/dx if y = (x^3 + 2x)^4.

Accepted Answer

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

Answer

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

Answer

(x^3 + 2x)^4

Answer

4(x^3 + 2x)^2

Question 12

Determine dy/dx if x = t^2 and y = sin(t).

Accepted Answer

2tcos(t) / t

Answer

tsin(t)

Answer

cos(t^2)

Answer

t^2cos(t)

Question 13

Find the derivative of f(x) = ln(x^2 + sqrt(x)).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 14

Find the derivative of f(x) = arctan(e^x).

Accepted Answer

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

Answer

e^x

Answer

arctan(e^x)

Answer

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

Question 15

Find the derivative of the function f(x) = (x^2 + 3x - 2)^5 using the Chain Rule.

Accepted Answer

10(x^2 + 3x - 2)^4(2x + 3)

Answer

10(x^2 + 3x - 2)^3

Answer

5(x^2 + 3x - 2)^4

Answer

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

Question 16

Find the derivative of the function h(x) = e^(2x^3 - 5) using the Chain Rule.

Accepted Answer

6x^2 e^(2x^3 - 5)

Answer

3x^2 e^(2x^3 - 5)

Answer

6xe^(2x^3 - 5)

Answer

3xe^(2x^3 - 5)

Question 17

Given the function g(x, y) = sin(x^2 + y^3), find the partial derivative ∂g/∂y.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 18

The volume of a sphere is given by V = (4/3)πr^3. Find the rate of change of the volume with respect to the radius.

Accepted Answer

4πr^2

Answer

πr

Answer

4r^2

Answer

πr^3

Question 19

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

cos(2x)

Answer

2x * sin(x^2 + 1)

Question 20

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

Accepted Answer

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

Answer

e^(cos(x))

Answer

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

Answer

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

Question 21

What is the derivative of y = ln(x^2 + 1)?

Accepted Answer

2x / (x^2 + 1)

Answer

2x

Answer

1 / (x^2 + 1)

Question 22

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

Accepted Answer

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

Answer

x / (x^2 + 1)

Answer

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

Answer

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

Question 23

If f(x) = g(h(x)), and g'(2) = 3, h'(1) = 4, and h(1) = 2, find f'(1).

Accepted Answer

12

Answer

4

Answer

3

Answer

7

Question 24

The position of a particle is given by s(t) = t^3 - 3t^2 + 2t. What is the acceleration of the particle at time t = 2?

Accepted Answer

6

Answer

2

Answer

0

Answer

12

Question 25

Find the derivative of f(x) = sin(cos(x)) using the Chain Rule.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 26

Determine the derivative of the function h(x) = sin(3x^2 + 2).

Accepted Answer

h'(x) = 6x * cos(3x^2 + 2)

Answer

h'(x) = (3x^2 + 2) * cos(3x^2 + 2)

Answer

h'(x) = 6x * sin(3x^2 + 2)

Answer

h'(x) = cos(3x^2 + 2)

Question 27

Find dy/dx if y = ln(sqrt(x^2 + 1)).

Accepted Answer

dy/dx = (x/(x^2 + 1)) * (1/2)

Answer

dy/dx = (x^2)/(2 * sqrt(x^2 + 1))

Answer

dy/dx = (x^2 + 1)/(2 * sqrt(x^2 + 1))

Answer

dy/dx = (1/sqrt(x^2 + 1))

Question 28

Using the chain rule, find the derivative of the function *h(x) = (sin(x))³*.

Accepted Answer

h'(x) = 3(sin(x))² * cos(x)

Answer

h'(x) = cos(x) * sin(x)³

Answer

h'(x) = 3sin(x)²

Answer

h'(x) = 3cos(x) * sin(x)²

Question 29

Determine the derivative of the function *f(x) = √(x³ + 2x)*.

Accepted Answer

f'(x) = (3x² + 2) / (2√(x³ + 2x))

Answer

f'(x) = (3x² + 2) * √(x³ + 2x)

Answer

f'(x) = (3x² + 2)

Answer

f'(x) = (x³ + 2x)^(-1/2)

Question 30

Find the derivative of the function *g(x) = ln(x² * e^x)*.

Accepted Answer

g'(x) = (2x + 1) / x

Answer

g'(x) = x² * e^x

Answer

g'(x) = (x² + e^x) / x

Answer

g'(x) = ln(x² * e^x)

Question 31

What is the derivative of *f(x) = e^(sin(x))*?

Accepted Answer

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

Answer

e^(sin(x))

Answer

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

Answer

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

Question 32

What is the derivative of *f(x) = (x² + 1)^(1/2)*?

Accepted Answer

x / (x² + 1)^(1/2)

Answer

x(x² + 1)^(-1/2)

Answer

x / (x² + 1)

Answer

1/2(x² + 1)^(-1/2)

Question 33

Find the derivative of *y = (x³ + 2x)⁵*.

Accepted Answer

5(x³ + 2x)⁴(3x² + 2)

Answer

(x³ + 2x)⁴(3x³ + 2x²)

Answer

(x³ + 2x)⁴(15x² + 10x)

Answer

5(x³ + 2x)⁵(3x² + 2)

Question 34

Find the derivative of the implicit function defined by the equation *y² = x³ + y*.

Accepted Answer

(3x² + 1) / (2y - 1)

Answer

(2y - 1) / (3x² + 1)

Answer

(3x² - 1) / (2y + 1)

Answer

(2y + 1) / (3x² - 1)

Question 35

Find the derivative of *f(x) = √(x² + 1) + tan⁻¹(x)*.

Accepted Answer

(x / √(x² + 1)) + 1 / (1 + x²)

Answer

(√(x² + 1) / x) + 1 / (1 + x²)

Answer

(√(x² + 1) / x) - 1 / (1 + x²)

Answer

(x / √(x² + 1)) - 1 / (1 + x²)

Question 36

What is the derivative of the function  f(x) = sin(x² + 1)?

Accepted Answer

2x cos(x² + 1)

Answer

cos(x² + 1)

Answer

2x sin(x² + 1)

Answer

cos(2x)

Question 37

What is the derivative of y = e^(3x² - 2x)?

Accepted Answer

(6x - 2)e^(3x² - 2x)

Answer

(3x² - 2x)e^(3x² - 2x)

Answer

e^(3x² - 2x)

Answer

(6x - 2)e^(6x - 2)

Question 38

Given u = x² + y² and v = xy, what is ∂u/∂x in terms of x and y?

Accepted Answer

2x

Answer

2x + y

Answer

2x² + y²

Answer

x + 2y

Question 39

A spherical balloon is being inflated at a rate of 10 cubic inches per second.  At the moment when the radius is 2 inches, how fast is the radius increasing?

Accepted Answer

5/4π inches per second

Answer

10/2π inches per second

Answer

5/2π inches per second

Answer

10/4π inches per second

Question 40

Using the Chain Rule, find the derivative of y = (x³ + 1)².

Accepted Answer

6x²(x³ + 1)

Answer

2(x³ + 1)

Answer

2x²(x³ + 1)

Answer

3x²(x³ + 1)

Question 41

Given the equation x² + y² = 25, what is dy/dx using implicit differentiation?

Accepted Answer

-x/y

Answer

y/x

Answer

x/y

Question 42

What is the derivative of f(x) = (sin(x))²?

Accepted Answer

2sin(x)cos(x)

Answer

2cos(x)sin(x)

Answer

cos(x)sin(x)

Question 43

Using the Chain Rule, what is the derivative of y = ln(x² + 1)?

Accepted Answer

2x/(x² + 1)

Answer

1/(x² + 1)

Answer

2x/(x² + 1)²

Answer

2x

Question 44

If z = f(u, v), where u = x² + y² and v = xy, find ∂z/∂x using the Chain Rule.

Accepted Answer

(∂f/∂u)(2x) + (∂f/∂v)(y)

Answer

(∂f/∂u)(x) + (∂f/∂v)(y)

Answer

(∂f/∂u)(2x) + (∂f/∂v)(x)

Question 45

A 10-foot ladder is leaning against a vertical wall.  The bottom of the ladder is sliding away from the wall at a rate of 1 foot per second.  When the bottom of the ladder is 6 feet from the wall, how fast is the top of the ladder sliding down the wall?

Accepted Answer

-3/4 feet per second

Answer

3/4 feet per second

Answer

-4/3 feet per second

Answer

4/3 feet per second