Differentiation in Calculus: Master Derivatives and Applications

Question 1

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

Accepted Answer

2 * e^(2x + 1)

Answer

e^(2x + 1)

Answer

1 / e^(2x + 1)

Answer

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

Question 2

Find the equation of the normal to the curve y = x² + 4 at the point (-2, 8).

Accepted Answer

y = 1/2x + 6

Answer

y = -1/2x + 6

Answer

y = -1/2x + 4

Answer

y = 1/2x + 4

Question 3

The position of a particle moving in a plane is given by r(t) = (t³ - 2t, t² + 1). What is the speed of the particle at time t = 1?

Accepted Answer

2√2

Answer

√2

Answer

2

Answer

4

Question 4

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

Accepted Answer

6x - 2

Answer

3x - 2

Answer

3x^2

Answer

2x - 1

Question 5

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

Accepted Answer

3e^(3x-1)

Answer

e^(3x-1)

Answer

3

Answer

1

Question 6

Find the local minimum point of the cubic function f(x) = x^3 - 3x^2 + 2.

Accepted Answer

(1, -2)

Answer

(2, -2)

Answer

(0, 2)

Answer

(3, 2)

Question 7

Find the rate of change of the function y = tan(x) at x = π/3.

Accepted Answer

√3

Answer

1

Answer

2

Answer

3

Question 8

Find the x-coordinate of the stationary point of the function h(x) = x^4 - 4x^2 + 3.

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 9

Determine the rate of change of the volume of a sphere with respect to its radius, given that V = (4/3)πr^3.

Accepted Answer

4πr^2

Answer

8πr^2

Answer

12πr^2

Answer

16πr^2

Question 10

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

Accepted Answer

2x/(x^2 + 1)

Answer

x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

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

Question 11

Determine the derivative of sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

cos(x)

Answer

-sin(2x)

Question 12

Which option is NOT a recognized differentiation rule?

Accepted Answer

Double derivative rule

Answer

Product rule

Answer

Quotient rule

Answer

Chain rule

Question 13

Determine the second derivative of the function g(x) = ln(x).

Accepted Answer

-1/x^2

Answer

1/x^2

Answer

-1/x

Answer

1/x

Question 14

Calculate the derivative of the function f(x) = ln(x² + 1).

Accepted Answer

2x/(x² + 1)

Answer

x/(x² + 1)

Answer

x²/(x² + 1)

Answer

1/(x² + 1)

Question 15

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

Accepted Answer

2x/(x² + 1)

Answer

x/(x² + 1)

Answer

1/(x² + 1)

Answer

2/(x² + 1)

Question 16

Which of the following functions has a constant derivative?

Accepted Answer

y = e^x

Answer

y = x²

Answer

y = sin(x)

Answer

y = ln(x)

Question 17

Find the stationary points of the function (f(x) = x^4 - 4x^3 + 3x^2).

Accepted Answer

x = 0, 2, 3

Answer

x = 0, 1, 3

Answer

x = 1, 2, 3

Answer

x = 0, 1, 2

Question 18

Find the rate of change of the volume of a sphere with respect to its radius.

Accepted Answer

dV/dr = 4πr^2

Answer

dV/dr = 2πr^2

Answer

dV/dr = πr^2

Answer

dV/dr = 4πr

Question 19

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

Accepted Answer

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

Answer

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

Answer

f'(x) = 2x

Answer

f'(x) = 1/x

Question 20

Find the value of the second derivative of the function (f(x) = e^(2x)) at (x = 0).

Accepted Answer

4

Answer

2

Answer

1

Answer

0

Question 21

Find the vertex (maximum point) of the parabola y = -x^2 + 4x.

Accepted Answer

(2, 4)

Answer

(0, 0)

Answer

(4, -8)

Answer

(-2, 4)

Question 22

What is the rate of change of the exponential function y = e^x?

Accepted Answer

y' = e^x

Answer

y' = x * e^x

Answer

y' = e^(-x)

Answer

y' = -x * e^x

Question 23

If f(x) is differentiable and f'(x) > 0 for all x, what can be concluded?

Accepted Answer

f(x) is increasing.

Answer

f(x) is decreasing.

Answer

f(x) is constant.

Answer

f(x) has a turning point.

Question 24

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

Accepted Answer

y'' = 4e^(2x + 1)

Answer

y'' = 2e^(2x + 1)

Answer

y'' = e^(2x + 1) / 2

Answer

y'' = e^(2x + 1)

Question 25

If the graph of a function has an inflection point at x = 2, what can be concluded?

Accepted Answer

f''(2) = 0

Answer

f'(2) = 0

Answer

f'(2) = ∞

Answer

f''(2) = ∞

Question 26

Find the slope of the tangent to the curve y = sin(x) at x = π/4. Express your answer as a simplified fraction or exact value.

Accepted Answer

√2 / 2

Answer

1

Answer

-1

Answer

√2 / 4

Question 27

Which of the following is the second derivative of f(x) = sin(3x)? Simplify your answer to the form Asin(Bx) + Ccos(Bx), where A, B, and C are constants.

Accepted Answer

-9cos(3x)

Answer

-9sin(3x)

Answer

9sin(3x)

Answer

9cos(3x)

Question 28

Which of the following is a technique used to find the derivative of an implicit function? Choose the most appropriate answer.

Accepted Answer

Implicit differentiation

Answer

Quotient rule

Answer

Product rule

Answer

Chain rule

Question 29

Determine the gradient of the tangent to the curve y = sin(x) at x = π/4.

Accepted Answer

√2

Answer

0

Answer

1

Answer

-√2

Question 30

Find both the local maximum and local minimum of the function f(x) = x^4 - 4x^2 + 3.

Accepted Answer

Local maximum: (0, 3); Local minimum: (2, -3)

Answer

Local maximum: (0, 3); Local minimum: none

Answer

Local maximum: none; Local minimum: (2, -3)

Answer

Local maximum: (2, -3); Local minimum: none

Question 31

Which of the following is NOT a central concept in the unit on differentiation?

Accepted Answer

Integration

Answer

Derivatives

Answer

Applications of derivatives

Answer

Extrema

Question 32

The derivative of a constant function is:

Accepted Answer

0

Answer

1

Answer

x

Answer

e

Question 33

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

Accepted Answer

2x - 3

Answer

x - 3

Answer

x² - 3

Answer

2x + 3

Question 34

Find the partial derivative of y with respect to x for the function f(x, y) = xy² + x²y.

Accepted Answer

y² + 2xy

Answer

xy + x²

Answer

y² - 2xy

Answer

xy - x²

Question 35

Calculate the second derivative of the function f(x) = ln(x).

Accepted Answer

-1/x²

Answer

1/x

Answer

-1/x

Answer

1/x²

Question 36

Determine the derivative of the function f(x) = |x - 1|.

Accepted Answer

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

Answer

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

Answer

1

Answer

|x + 1|

Question 37

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

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-cos(x)

Answer

-sin(x)

Question 38

Determine the gradient of the normal to the curve y = x^2 at the point (2, 4).

Accepted Answer

1/2

Answer

-1/2

Answer

2

Answer

-2

Question 39

Find the equation of the tangent to the curve y = e^x at the point (0, 1).

Accepted Answer

y = x + 1

Answer

y = e^x

Answer

y = x - 1

Answer

y = -x + 1

Question 40

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

Accepted Answer

1/x^2

Answer

-1/x^2

Answer

1/x

Answer

-1/x

Question 41

What is the rate of change of the function f(x) = 2x^3 + 5x^2 - 10x + 4 at x = 2?

Accepted Answer

32

Answer

28

Answer

36

Answer

40

Question 42

Find the maximum value of the function f(x) = -x^2 + 4x + 3.

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 43

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

Accepted Answer

3x² - 4x

Answer

x² - 4x

Answer

3x² + 4x

Answer

x² + 4x

Question 44

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

Accepted Answer

e^x

Answer

e

Answer

x^e

Answer

e*x

Question 45

Identify the critical points of the function f(x) = x³ - 6x² + 9x.

Accepted Answer

x = 0, 3

Answer

x = 0, 1

Answer

x = 1, 3

Answer

x = 2, 3

Question 46

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

Accepted Answer

6x^2 - 10x + 3

Answer

6x^2 + 10x - 3

Answer

6x^2 + 10x + 3

Answer

6x^2 - 10x - 3

Question 47

Calculate the gradient of the curve y = sin(x) at the point x = π/4.

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

-√2/2

Question 48

Determine the rate of change of the surface area of a sphere with respect to its radius.

Accepted Answer

4πr

Answer

πr

Answer

2πr

Answer

4πr^2

Question 49

Which of the following is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

e^x

Question 50

Which of the following functions has an inflection point at x = 0?

Accepted Answer

f(x) = x^4

Answer

f(x) = x^3

Answer

f(x) = x^5

Answer

f(x) = x^6

Question 51

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

Accepted Answer

2sec^2(2x)

Answer

2tan^2(2x)

Answer

2cosec^2(2x)

Answer

2cot^2(2x)

Question 52

Calculate the derivative of x^3.

Accepted Answer

3x^2

Answer

x^2

Answer

3x

Answer

x

Question 53

Find the derivative of the inverse function of y = e^x.

Accepted Answer

1/ln(x)

Answer

e^-x

Answer

1/e^x

Answer

ln(x)