Calculus Differentiation: Master Derivatives with Interactive Quizzes

Question 1

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-sin(2x)

Question 2

Determine the derivative of the exponential function f(x) = e^x.

Accepted Answer

e^x

Answer

x * e^x

Answer

e^-x

Answer

-e^x

Question 3

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

Accepted Answer

2x / x^2

Answer

2 / x

Answer

-2 / x^2

Answer

2x / x

Question 4

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

Accepted Answer

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

Answer

x^x * ln(x)

Answer

(x^x) * ln(x)

Answer

x^(x-1) * ln(x)

Question 5

Find the derivative of y = sin(2x):

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-cos(2x)

Question 6

Calculate the derivative of f(x) = e^x:

Accepted Answer

e^x

Answer

x * e^x

Answer

-e^x

Answer

1/e^x

Question 7

Find the derivative of f(x) = ln(x² + 1):

Accepted Answer

2x/(x² + 1)

Answer

1/(x² + 1)

Answer

2x

Answer

1/x

Question 8

Calculate the derivative of f(x) = √x + 1/√x:

Accepted Answer

1/2√x + 1/2√x

Answer

1/2√x - 1/2√x

Answer

-1/2√x - 1/2√x

Answer

-1/2√x + 1/2√x

Question 9

Find the derivative of f(x) = sin(x² + 1):

Accepted Answer

2xcos(x² + 1)

Answer

xcos(x² + 1)

Answer

-2xcos(x² + 1)

Answer

-xcos(x² + 1)

Question 10

Which of the following is the derivative of the function f(x) = x^3 - 2x^2 + 5x + 1?

Accepted Answer

3x^2 - 4x + 5

Answer

x^2 - 4x + 5

Answer

3x^2 - 2x + 5

Answer

x^2 - 2x + 5

Question 11

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

Accepted Answer

2cos(2x + 1)

Answer

cos(2x + 1)

Answer

-2cos(2x + 1)

Answer

-cos(2x + 1)

Question 12

Which of the following is the derivative of the function f(x) = ln(x^2 + 1)?

Accepted Answer

2x/(x^2 + 1)

Answer

x/(x^2 + 1)

Answer

-2x/(x^2 + 1)

Answer

-x/(x^2 + 1)

Question 13

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

Accepted Answer

3e^(3x - 1)

Answer

3xe^(3x - 1)

Answer

e^(3x - 1)

Answer

-e^(3x-1) / 3

Question 14

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

Calculate the derivative of the function f(x) = 3x² + 2x - 1.

Accepted Answer

6x + 2

Answer

9x² + 2

Answer

3x + 2

Answer

6x - 1

Question 16

Find the derivative of the function g(x) = sin(x) + cos(x).

Accepted Answer

cos(x) - sin(x)

Answer

-cos(x) - sin(x)

Answer

sin(x) - cos(x)

Answer

-sin(x) + cos(x)

Question 17

Which statement accurately describes the function f(x) = |x|?

Accepted Answer

Continuous but not differentiable at x = 0.

Answer

Differentiable at all points.

Answer

Discontinuous at all points.

Answer

Not continuous at x = 0.

Question 18

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

Accepted Answer

6x - 2

Answer

6x^2 - 2

Answer

3x - 2

Answer

9x - 2

Question 19

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

Accepted Answer

2cos(2x)

Answer

2sin(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Question 20

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

Accepted Answer

3e^(3x)

Answer

3x * e^(3x)

Answer

e^(3x)

Answer

e^(3x)/3

Question 21

At the point (1, -1) on the curve y = x^3 - 2x, calculate the gradient of the tangent line.

Accepted Answer

1

Answer

-3

Answer

-1

Answer

3

Question 22

A particle's position is given by the function s(t) = t^2 - 4t. Determine the particle's velocity at time t = 2.

Accepted Answer

0

Answer

-4

Answer

4

Answer

2

Question 23

Find the second derivative of the function f(x) = x^4 + 2x^3.

Accepted Answer

12x^2 + 12x

Answer

4x^2 + 6x

Answer

4x^3 + 6x^2

Answer

24x + 12

Question 24

Determine the critical points of the function f(x) = x^3 - 3x^2.

Accepted Answer

x = 0 and x = 2

Answer

x = 1 and x = 3

Answer

x = 2 and x = 3

Answer

x = 0 and x = 1

Question 25

Which of the following functions has a derivative of f'(x) = 2x?

Accepted Answer

f(x) = x^2

Answer

f(x) = 2x^2

Answer

f(x) = x^2 + 1

Answer

f(x) = x^2 - 1

Question 26

The function f(x) = x^4 - 2x^3 is increasing on which interval?

Accepted Answer

(1.5, ∞)

Answer

(0, 1.5)

Answer

(0, 1)

Answer

(-∞, 0)

Question 27

The graph of a function has a horizontal tangent at x = 2. What can be said about the derivative of the function at x = 2?

Accepted Answer

The derivative is equal to zero.

Answer

The derivative does not exist.

Answer

The derivative is negative.

Answer

The derivative is positive.