Master the Derivative: Enhance Your Calculus Skills

Question 1

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

x*cos(x)

Answer

1

Question 2

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

Accepted Answer

e^x

Answer

x*e^x

Answer

1

Answer

0

Question 3

Which of the following represents the chain rule of differentiation?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

Determine the derivative of the function f(x) = x³ + 2x² - 5.

Accepted Answer

3x² + 4x

Answer

x² + 4x

Answer

3x² - 4x

Answer

x² - 4x

Question 5

Apply the chain rule to find the derivative of the function f(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-cos(2x)

Answer

-2cos(2x)

Question 6

Find the second derivative of the function f(x) = e^(-x²).

Accepted Answer

2(x² - 1)e^(-x²)

Answer

-2xe^(-x²)

Answer

2xe^(-x²)

Answer

-2(x² - 1)e^(-x²)

Question 7

Which of the following is NOT an application of derivatives in real-world scenarios?

Accepted Answer

Predicting weather patterns

Answer

Finding slopes of curves

Answer

Optimizing functions

Answer

Solving differential equations

Question 8

Find the slope of the tangent line to the curve y = x^2 at the point (1, 1).

Accepted Answer

2

Answer

1

Answer

-1

Answer

0

Question 9

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

Accepted Answer

2x/(x^2 + 1)

Answer

(x^2 + 1)/2x

Answer

2xln(x^2 + 1)

Answer

ln(x^2 + 1)/2x

Question 10

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

Accepted Answer

y = 2x - 3

Answer

y = x - 1

Answer

y = 3x - 4

Answer

y = -x + 1

Question 11

Calculate the derivative of the function f(x) = sqrt(x).

Accepted Answer

1/2 * sqrt(x)

Answer

1/sqrt(x)

Answer

-1/2 * sqrt(x)

Answer

sqrt(x)

Question 12

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

Accepted Answer

x^sin(x) * (sin(x) + xcos(x))

Answer

sin(x) * ln(x)

Answer

x^cos(x)

Answer

cos(x) * ln(x)

Question 13

Using the definition of the derivative, prove the power rule:  If  *f(x) = x^n*, then *f'(x) = nx^(n-1)*.

Accepted Answer

f'(x) = nx^(n-1)

Answer

f'(x) = n!x^(n-1)

Answer

f'(x) = nx^n

Answer

f'(x) = nx^(n-1) + C

Question 14

Find the definition of the derivative of a function f(x).

Accepted Answer

The instantaneous rate of change of f(x) at x.

Answer

The line that just touches the graph of f(x) at x.

Answer

The area under the curve of f(x) at x.

Answer

The integral of f(x) at x.

Question 15

Find the derivative of f(x) = ln(x).

Accepted Answer

1 / x

Answer

x

Answer

-1 / x

Answer

ln(x)

Question 16

Find the derivative of the function h(x) = e^(2x) with respect to x.

Accepted Answer

h'(x) = 2e^(2x)

Answer

h'(x) = e^(2x/2)

Answer

h'(x) = e^(2x)/2

Answer

h'(x) = 2xe^(2x)

Question 17

Find the derivative of the function f(x) = ln(x) with respect to x.

Accepted Answer

f'(x) = 1/x

Answer

f'(x) = ln(x)

Answer

f'(x) = -ln(x)

Answer

f'(x) = x

Question 18

Find the derivative of the function f(x) = sqrt(x) with respect to x.

Accepted Answer

f'(x) = 1/(2sqrt(x))

Answer

f'(x) = 2sqrt(x)

Answer

f'(x) = sqrt(x)/2

Answer

f'(x) = 1/sqrt(x)

Question 19

Find the derivative of the function f(x) = cot(x) with respect to x.

Accepted Answer

f'(x) = -csc^2(x)

Answer

f'(x) = cot(x)

Answer

f'(x) = sec(x)

Answer

f'(x) = csc(x)

Question 20

What is the geometric interpretation of the derivative?

Accepted Answer

The slope of the tangent line to a function at a given point

Answer

The average rate of change of the function over an interval

Answer

The rate at which the function is increasing or decreasing

Question 21

Find the derivative of the function y = x^3.

Accepted Answer

y' = 3x^2

Answer

y' = x

Answer

y' = x^2

Answer

y' = 3x

Question 22

Calculate the derivative of f(x) = sin(x).

Accepted Answer

f'(x) = cos(x)

Answer

f'(x) = -sin(x)

Answer

f'(x) = x

Answer

f'(x) = tan(x)

Question 23

Determine the derivative of g(x) = e^x.

Accepted Answer

g'(x) = e^x

Answer

g'(x) = 0

Answer

g'(x) = e

Answer

g'(x) = x * e^x

Question 24

Find the derivative of h(x) = ln(x).

Accepted Answer

h'(x) = 1/x

Answer

h'(x) = x

Answer

h'(x) = -1/x

Answer

h'(x) = 0

Question 25

Find the derivative of f(x) = |x|.

Accepted Answer

f'(x) = 1 for x > 0, f'(x) = -1 for x < 0

Answer

f'(x) = 0

Answer

f'(x) = -1

Answer

f'(x) = 1

Question 26

Which of the following functions has a positive derivative for all positive values of x?

Accepted Answer

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

Answer

f(x) = x^2 - 4x

Answer

f(x) = |x|

Answer

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

Question 27

Identify the critical points of the function g(x) = x^4 - 2x^2.

Accepted Answer

x = 0, x = ±√2

Answer

x = 1, x = -1

Answer

x = ±2

Answer

x = 0, x = ±1

Question 28

Given the function f(x) = x^3 + 2x^2 - 1, calculate its derivative.

Accepted Answer

3x^2 + 4x

Answer

3x^4 + 4x^3

Answer

x^2 + 2x

Question 29

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

Accepted Answer

2e^(2x)

Answer

4xe^(2x)

Answer

e^(2x)

Answer

-2xe^(2x)

Question 30

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

Accepted Answer

2x*cos(x^2)

Answer

cos(x^2)

Answer

2*sin(x^2)

Answer

x^2*cos(x^2)

Question 31

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

Accepted Answer

1/(1+x^2)

Answer

arctan(x^2)

Answer

-1/(1+x^2)

Answer

x/(1+x^2)

Question 32

Which statement accurately describes a derivative?

Accepted Answer

The derivative of a function measures its instantaneous rate of change with respect to its independent variable.

Answer

The derivative represents the average rate of change of a function over an interval.

Answer

The derivative is the area under the curve of a function.

Answer

The derivative is the slope of the secant line to a graph at a given point.

Question 33

What is the derivative of the function *f(x) = x²*?

Accepted Answer

2x

Answer

x³

Answer

x

Answer

½x²

Question 34

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

Accepted Answer

e^x

Answer

e

Answer

x^e

Answer

0

Question 35

What is the derivative of the function *f(x) = ln(x)*?

Accepted Answer

1/x

Answer

0

Answer

x

Answer

e^x

Question 36

What is the derivative of the function *f(x) = |x|*?

Accepted Answer

1 for x > 0 and -1 for x < 0

Answer

1 for all x

Answer

-1 for all x

Answer

0 for all x

Question 37

What is the derivative of the function *f(x) = g(h(x))*?

Accepted Answer

f'(x) = g'(h(x)) * h'(x)

Answer

f'(x) = g'(x) / h'(x)

Answer

f'(x) = h'(g(x))

Answer

f'(x) = g'(x) + h'(x)

Question 38

What is the derivative of the function  *f(x) = x³ + 2x² - 5x + 1*?

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 5x

Answer

3x³ + 4x² - 5

Answer

x² + 4x - 5

Question 39

What is the derivative of the function *y = sin(x)*?

Accepted Answer

cos(x)

Answer

xcos(x)

Answer

sin(x)

Answer

-sin(x)

Question 40

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

Accepted Answer

e^x

Answer

e

Answer

x^e

Answer

xe^x

Question 41

What is the derivative of the logarithmic function *f(x) = ln(x)*?

Accepted Answer

1/x

Answer

ln(x)

Answer

-1/x

Answer

x

Question 42

What is the derivative of the function *f(x) = |x|*?

Accepted Answer

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

Answer

0

Answer

1

Answer

x

Question 43

What is the derivative of the function *f(x) = arctan(x)*?

Accepted Answer

1/(1 + x²)

Answer

-1/(1 + x²)

Answer

x

Answer

arctan(x)

Question 44

What is the derivative of the function *f(x) = √x*?

Accepted Answer

1/(2√x)

Answer

2√x

Answer

1/√x

Question 45

What is the derivative of the function *f(x) = (x² + 1)⁵*?

Accepted Answer

10x(x² + 1)⁴

Answer

(x² + 1)⁵

Answer

x(x² + 1)⁵

Answer

5(x² + 1)⁴

Question 46

What is the derivative of the function *f(x) = e^(x²)*?

Accepted Answer

2xe^(x²)

Answer

e^(x²)

Answer

-2xe^(x²)

Answer

x²e^(x²)