Master Differentiation in Calculus: Interactive Quizzes and Practice

Question 1

Which of the following correctly defines the power rule of differentiation?

Accepted Answer

The derivative of x^n with respect to x is nx^(n-1).

Answer

The derivative of sin(x) with respect to x is cos(x).

Answer

The derivative of e^x with respect to x is e^x.

Answer

The derivative of ln(x) with respect to x is 1/x.

Question 2

Find the derivative of the function f(x) = sin(3x) using the chain rule.

Accepted Answer

3cos(3x)

Answer

cos(3x)

Answer

1/3cos(3x)

Answer

3sin(3x)

Question 3

Identify the function with a zero derivative.

Accepted Answer

y = constant

Answer

y = x^2

Answer

y = e^x

Answer

y = sin(x)

Question 4

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

Accepted Answer

y = -x - 1

Answer

y = -x + 1

Answer

y = x - 1

Answer

y = x + 1

Question 5

Derive the power rule of differentiation: $$rac{d(x^n)}{dx}=nx^{n-1}$$. Explain the steps involved in the derivation and provide a real-world example to illustrate its application.

Accepted Answer

Use the limit definition of a derivative to show that $$lim_{h 	o 0} rac{(x+h)^n - x^n}{h} = nx^{n-1}$$

Answer

Apply the product rule to $$(x^n)'$$

Answer

Use the quotient rule to $$rac{d(x^n)}{dx}$$

Answer

The power rule is a fundamental theorem that cannot be derived

Question 6

Find the derivative of the function **f(x) = x^3 + 2x^2 - 5x + 1** using the power rule.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 - 4x - 5

Question 7

Find the derivative of the function **f(x) = (x^2 + 1)(x - 2)** using the product rule.

Accepted Answer

2x^3 - 6x^2 + 4x

Answer

x^3 - 6x^2 + 4x

Answer

2x^3 + 6x^2 + 4x

Question 8

Find the derivative of the function **f(x) = x/(x^2 + 1)** using the quotient rule.

Accepted Answer

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

Answer

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

Answer

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

Question 9

Which of the following is the derivative of **f(x) = e^(x^3 - 2x)**?

Accepted Answer

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

Answer

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

Answer

3x^2 * e^(-x^3 + 2x) + 2 * e^(-x^3 + 2x)

Question 10

Find the derivative of the function f(x) = x³ + 2x² - 5x + 1:

Accepted Answer

f'(x) = 3x² + 4x - 5

Answer

f'(x) = x³ + 4x² - 5x

Answer

f'(x) = 3x² - 4x + 5

Question 11

Find the derivative of the implicit function defined by the equation x² + y² = 25:

Accepted Answer

dy/dx = -x/y

Answer

dy/dx = y/x

Answer

dy/dx = x/y

Answer

dy/dx = 0

Question 12

Find the critical points of the function f(x) = x⁴ - 4x² + 3:

Accepted Answer

x = ±1, ±√3

Answer

x = ±3, ±√5

Answer

x = 0, ±1

Answer

x = ±2, ±√3

Question 13

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

Accepted Answer

h'(x) = (1 - x^2)/(x^2 + 1)^2

Answer

h'(x) = 1/(x^2 + 1)

Answer

h'(x) = x^2/(x^2 + 1)

Answer

h'(x) = (x^2 + 1)/x

Question 14

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

Accepted Answer

f'(x) = 2cos(2x)

Answer

f'(x) = sin(2x)

Answer

f'(x) = cos(2x)

Question 15

Find the derivative of the exponential function f(x) = e^(3x + 1).

Accepted Answer

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

Answer

f'(x) = 3x*e^(3x + 1)

Answer

f'(x) = (3x + 1)*e^(3x + 1)

Answer

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

Question 16

Determine the derivative of the logarithmic function f(x) = ln(x^2 - 1).

Accepted Answer

f'(x) = 2x/(x^2 - 1)

Answer

f'(x) = ln(x^2 - 1)

Answer

f'(x) = 2x

Answer

f'(x) = 1/(x^2 - 1)

Question 17

Calculate the derivative of the function f(x) = x^2 * e^(-x).

Accepted Answer

f'(x) = x^2 * e^(-x) - 2x * e^(-x)

Answer

f'(x) = x^2 * e^(-2x)

Answer

f'(x) = x^2 * e^(x)

Answer

f'(x) = 2x * e^(-x)

Question 18

Find the derivative of the function f(x) = (sin(x) + cos(x))^2.

Accepted Answer

f'(x) = 2(sin(x) + cos(x)) * (cos(x) - sin(x))

Answer

f'(x) = 2(sin(x) - cos(x)) * (cos(x) + sin(x))

Answer

f'(x) = (sin(x) + cos(x)) * (cos(x) - sin(x))

Answer

f'(x) = 2(sin(x) - cos(x)) * (sin(x) - cos(x))

Question 19

Calculate the derivative of f(x) = (2x + 1)(x - 3). Which rule of differentiation should be used to find the derivative of this function?

Accepted Answer

Product rule

Answer

Power rule

Answer

Quotient rule

Answer

Chain rule

Question 20

Find the derivative of f(x) = x/(x + 1). Which rule of differentiation is used to differentiate this function?

Accepted Answer

Quotient rule

Answer

Power rule

Answer

Product rule

Answer

Chain rule

Question 21

What is the derivative of f(x) = sin(x²)? Identify the rule that must be used to find the derivative of this composed function.

Accepted Answer

Chain rule

Answer

Quotient rule

Answer

Product rule

Answer

Power rule

Question 22

Calculate the derivative of f(x) = ln(x). Which logarithmic differentiation rule is used to find the derivative of ln(x)?

Accepted Answer

d/dx(ln(x)) = 1/x

Answer

d/dx(ln(x)) = e^x

Answer

d/dx(ln(x)) = x

Answer

d/dx(ln(x)) = ln(x^2)

Question 23

Find the derivative of the function f(x) = sqrt(x). Apply the power rule to differentiate this rational function, and what value should be used for the exponent?

Accepted Answer

Exponent: 1/2

Answer

Exponent: x

Answer

Exponent: 2

Answer

Exponent: 1

Question 24

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

Accepted Answer

2x + 2

Answer

x^2

Answer

2x - 1

Answer

1

Question 25

Find the derivative of the function g(x) = (x^3 - 2x^2 + 1) * (x + 1).

Accepted Answer

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

Answer

3x^2 - 4x + 1

Answer

x^3 - 2x^2 + 1

Answer

x^4 - 2x^3 + x + 1

Question 26

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

Accepted Answer

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

Answer

1 / (x - 1)

Answer

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

Answer

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

Question 27

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

Accepted Answer

2e^(2x)

Answer

e^2 * x

Answer

e^(x^2)

Answer

2x * e^(2x)

Question 28

Find the derivative of the function g(x) = ln(3x + 2).

Accepted Answer

3/(3x + 2)

Answer

ln(3) / (3x + 2)

Answer

1/(3x + 2)

Question 29

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

Accepted Answer

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

Answer

2tan(x)

Answer

sec^2(x)

Answer

tan^2(x)

Question 30

Find the second derivative of the function g(x) = ln(x^2 + x + 1).

Accepted Answer

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

Answer

1 / (x^2 + x + 1)

Answer

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

Question 31

What is the derivative of the function  *f(x) = 3x⁴ - 2x² + 5*?

Accepted Answer

12x³ - 4x

Answer

12x⁴ - 4x²

Answer

12x³ - 4x + 5

Answer

4x³ - 4x

Question 32

Find the derivative of *h(x) = sin(2x + 1)* using the chain rule.

Accepted Answer

2cos(2x + 1)

Answer

-2cos(2x + 1)

Answer

2sin(2x + 1)

Answer

cos(2x + 1)

Question 33

Calculate the second derivative of the function *y = e^x * sin(x)*.

Accepted Answer

2e^x * cos(x)

Answer

2e^x * sin(x)

Answer

e^x * cos(x)

Answer

-2e^x * sin(x)

Question 34

Which of the following functions has a derivative that is always positive?

Accepted Answer

f(x) = e^x

Answer

g(x) = x²

Answer

k(x) = ln(x)

Answer

h(x) = sin(x)

Question 35

What is the derivative of the function  *f(x) = x² + 2x - 1*  using the power rule?

Accepted Answer

2x + 2

Answer

x² + 2

Answer

4x - 1

Answer

x³ + 2x² - x

Question 36

Use the quotient rule to find the derivative of the function  *g(x) = (x³ - 2x + 1) / (x² + 1)*.

Accepted Answer

(x⁴ - 2x³ - 4x² + 2x - 1) / (x² + 1)²

Answer

((x⁴ - 2x³ + 1)/(x² + 1)) * (x² - 2)

Answer

((x³ - 2x + 1)/(x² + 1)) * (2x - 1)

Answer

((x³ - 2x + 1)²)/(x² + 1)³

Question 37

Apply the chain rule to find the derivative of the function *h(x) = ln(x² - 1)*.

Accepted Answer

(2x) / (x² - 1)

Answer

(ln(x² - 1)) / (2x)

Answer

2x * ln(x² - 1)

Answer

1 / (x² - 1)

Question 38

What is the derivative of *f(x) = e^(x² + 1)*  using the chain rule?

Accepted Answer

2xe^(x² + 1)

Answer

x * e^(x² + 1)

Answer

e^(x² + 1)

Answer

(x² + 1) * e^(x²)

Question 39

Find the derivative of the function *g(x) = sin(2x + 1)*  using the chain rule.

Accepted Answer

2cos(2x + 1)

Answer

cos(2x + 1)

Answer

sin(2x)

Answer

-2sin(2x + 1)

Question 40

What is the derivative of *f(x) = √(x³ + 2x)*  using the chain rule?

Accepted Answer

(3x² + 2) / (2√(x³ + 2x))

Answer

1 / (2√(x³ + 2x))

Answer

3x² * √(x³ + 2x)

Answer

(x³ + 2x)^(1/2)

Question 41

Find the derivative of *g(x) = (x² + 1)³* using the chain rule.

Accepted Answer

6x(x² + 1)²

Answer

3x(x² + 1)²

Answer

2x / (x² + 1)³

Answer

(x² + 1)³

Question 42

Apply the chain rule to find the derivative of the function *f(x) = arctan(x² + 1)*.

Accepted Answer

2x / (1 + x⁴)

Answer

arctan(x² + 1)

Answer

-2x / (1 + x⁴)

Answer

1 / (1 + x⁴)

Question 43

What is the derivative of *g(x) = ln(e^x + x²)* using the chain rule?

Accepted Answer

(e^x + 2x) / (e^x + x²)

Answer

1 / (e^x + x²)

Answer

(e^x - x²) / (e^x + x²)

Answer

e^x * ln(e^x + x²)

Question 44

Use the product rule to find the derivative of the function *f(x) = sin(x) * cos(x)*.

Accepted Answer

cos²(x) - sin²(x)

Answer

-sin(x) * cos(x)

Answer

sin(x) - cos(x)

Answer

sin(x) * cos(x)