Master the Derivative: Enhance Your Calculus II Skills

Question 1

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

Accepted Answer

e^x + 1/x

Answer

e^x - 1/x

Answer

e^x / x

Answer

e^x + ln(x)

Question 2

Use the limit definition of the derivative to find the derivative of the function f(x) = x^2.

Accepted Answer

2x

Answer

x

Answer

x^2

Answer

1

Question 3

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

Accepted Answer

5(x^2 + 1)^4 * 2x

Answer

10(x^2 + 1)^5

Answer

2x(x^2 + 1)^4

Answer

5x(x^2 + 1)^4

Question 4

Which of the following is a real-world application of the derivative in optimization?

Accepted Answer

Finding maximum and minimum values of functions

Answer

Solving trigonometric equations

Answer

Evaluating integrals

Answer

Calculating areas

Question 5

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

Accepted Answer

2x cos(x² + 1)

Answer

cos(x² + 1)

Answer

x cos(x² + 1)

Answer

-2x cos(x² + 1)

Question 6

Find the derivative of the function g(x) = |x - 3|.

Accepted Answer

1 if x > 3, -1 if x < 3

Answer

1 if x < 3, -1 if x > 3

Answer

1 for all x values

Answer

-1 for all x values

Question 7

If f(x) is differentiable and f'(x) is positive for all x, then f(x) is:

Accepted Answer

Increasing on the entire real line.

Answer

Decreasing on the entire real line.

Answer

Concave up on the entire real line.

Answer

Concave down on the entire real line.

Question 8

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

Accepted Answer

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

Answer

5(x^2 - 3)^4

Answer

(x^2 - 3)^4 * 2x

Answer

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

Question 9

Find the absolute maximum of the function f(x) = 2x^3 - 3x^2 - 12x + 5 on the interval [-2, 3].

Accepted Answer

22

Answer

13

Answer

5

Answer

-5

Question 10

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

Accepted Answer

2x/(x^2 + 1)

Answer

x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

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

Question 11

Calculate the derivative of f(x) = 2^x + log_2(x).

Accepted Answer

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

Answer

2^x + 1/x

Answer

2^x - 1/x

Answer

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

Question 12

Which of the following is NOT a rule for differentiating trigonometric functions?

Accepted Answer

d/dx(cot(x)) = -csc^2(x)

Answer

d/dx(sin(x)) = cos(x)

Answer

d/dx(tan(x)) = sec^2(x)

Answer

d/dx(cos(x)) = -sin(x)

Question 13

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

Accepted Answer

2x*cos(x^2)

Answer

cos(x^2)

Answer

x*cos(x^2)

Answer

x^2*cos(x^2)

Question 14

Determine the derivative of the exponential function f(x) = e^(3x - 2).

Accepted Answer

3e^(3x - 2)

Answer

-2e^(3x - 2)

Answer

e^(3x - 2)

Answer

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

Question 15

If f(x) = ln(x^2 + 1), calculate f'(x).

Accepted Answer

2x/(x^2 + 1)

Answer

x^2/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

ln(x^2 + 1)

Question 16

Calculate the derivative of y = ln(x² + 1).

Accepted Answer

2x / (x² + 1)

Answer

(x² + 1) / 2x

Answer

1 / (x² + 1)

Answer

x²

Question 17

Find the second derivative of f(x) = e^(3x - 2).

Accepted Answer

9e^(3x - 2)

Answer

3e^(3x - 2)

Answer

0

Answer

e^(3x - 2)

Question 18

Apply the chain rule to find the derivative of f(x) = (sin(x))³. Use clear and simplified language.

Accepted Answer

3(sin(x))² * cos(x)

Answer

(sin(x))³ * cos(x)

Answer

cos(x)

Answer

(sin(x))² * cos(x)

Question 19

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

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-sin(x)

Answer

cos(2x)

Question 20

Find the derivative of f(x) = e^(2x). Ensure the language is clear and succinct.

Accepted Answer

2e^(2x)

Answer

e^(2)

Answer

2x.e^(2x)

Question 21

Find the derivative of f(x) = ln(x²). Use simplified and clear language.

Accepted Answer

2x/x² = 2/x

Answer

ln(2x)

Answer

x²/x = x

Answer

1/x²

Question 22

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

Accepted Answer

2cos(2x + 1)

Answer

sin(2x + 1)

Answer

cos(2x + 1)

Answer

-2cos(2x + 1)

Question 23

Which of the following is NOT a fundamental technique used in differentiation?

Accepted Answer

Integration by parts

Answer

Chain rule

Answer

Product rule

Answer

Power rule

Question 24

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

Accepted Answer

2x * cos(x^2)

Answer

sin(2x)

Answer

x * cos(x^2)

Answer

cos(x^2)

Question 25

Calculate the derivative of the function g(x) = ln(x^3 + e^x).

Accepted Answer

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

Answer

x^3 + e^x

Answer

3x^2 + e^x

Answer

x^2 / e^x

Question 26

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

Accepted Answer

2^x * ln(2)

Answer

ln(2^x)

Answer

2x * 2^(x-1)

Answer

2^x * log(2)

Question 27

Calculate the derivative of the function g(x) = e^(x^2 + ln(x)) using implicit differentiation.

Accepted Answer

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

Answer

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

Answer

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

Answer

e^(x^2 + ln(x))

Question 28

Find the derivative of the function f(x) = sin(2x + 3) using the chain rule.

Accepted Answer

2cos(2x + 3)

Answer

cos(2x + 3)

Answer

sin(x + 3)

Answer

-2sin(2x + 3)

Question 29

Calculate the derivative of the function g(x) = e^(x^2 - 1) using the chain rule.

Accepted Answer

2xe^(x^2 - 1)

Answer

e^(x^2 - 1)

Answer

(x^2 - 1)e^(x^2 - 1)

Answer

x^2e^(x^2 - 1)

Question 30

Find the derivative of the function h(x) = ln(x^2 + 5) using the chain rule.

Accepted Answer

2x/(x^2 + 5)

Answer

x/(2x^2 + 10)

Answer

1/(x^2 + 5)

Answer

(x^2 + 5)/x

Question 31

Determine the derivative of the function f(x) = (x^3 + 2x^2 - 5)/(x^2 - 1) using the quotient rule.

Accepted Answer

(x^4 - 2x^3 + 13x - 5)/(x^2 - 1)^2

Answer

1

Answer

(3x^2 + 4x - 5)/(x^2 - 1)^2

Answer

x^3 + 2x^2 - 5

Question 32

Find the derivative of the function g(x) = sqrt(x^3 - 2x) using the chain rule.

Accepted Answer

(3x^2 - 2)/2sqrt(x^3 - 2x)

Answer

(3x^2 - 2)

Answer

sqrt(x^3 - 2x)

Answer

(3x^2 + 2)/2sqrt(x^3 - 2x)

Question 33

Determine the derivative of the function g(x) = (sin(x) + cos(x))^2 using the chain rule and product rule.

Accepted Answer

2(sin(x) + cos(x))(cos(x) - sin(x))

Answer

0

Answer

2(sin(x) - cos(x))

Answer

(sin(x) + cos(x))

Question 34

Find the derivative of the function h(x) = e^(x) * ln(x) using the product rule and logarithmic differentiation.

Accepted Answer

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

Answer

e^(x) * ln(x)

Answer

e^(x)