Calculus Derivative Questions with Solutions

This page presents a set of questions on the derivative of a function. Each question includes a detailed solution to help you gain a solid understanding of derivatives, a core concept in calculus.

Questions with Solutions

Question 1

If functions \( f \) and \( g \) satisfy

\( f(x) = g(x) + k \) where \( k \) is a constant, then:
(A) \( f'(x) = g'(x) + k \)
(B) \( f'(x) = g'(x) \)
(C) None of the above
Answer: (B). The derivative of a constant is zero, so \( f'(x) = g'(x) \).

Question 2

If \( f(x) = g(u) \) and \( u = u(x) \), then:
(A) \( f'(x) = g'(u) \)
(B) \( f'(x) = g'(u) \cdot u'(x) \)
(C) \( f'(x) = u'(x) \)
(D) None of the above
Answer: (B). This is the chain rule for the derivative of a composition of functions.

Question 3

Compute:

\( \lim_{x \to 0} \frac{e^x - 1}{x} \)
(A) 1
(B) 0
(C) Cannot be calculated (0/0 form)
Answer: (A). By definition of the derivative:
\( f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \)
For \( f(x) = e^x \), \( f'(x) = e^x \). Evaluating at \( x=0 \) gives \( e^0 = 1 \).

Question 4

True or False: The derivative of \( [g(x)]^2 \) is \( [g'(x)]^2 \).
Answer: False. The derivative of \( [g(x)]^2 \) is \( 2 g(x) g'(x) \).

Question 5

True or False: The derivative of \( f(x) \cdot g(x) \) is \( f'(x) \cdot g(x) + f(x) \cdot g'(x) \).
Answer: True.

Question 6

If \( f'(0) = 2 \), \( f'(2) = -3 \), \( f'(5) = 7 \), then

\( \lim_{x \to 4} \frac{f(x) - f(4)}{x - 4} \)
(A) 2
(B) -3
(C) 7
(D) None of the above
Answer: (D). This limit equals \( f'(4) \) by definition of the derivative.

Question 7

If \( f'(x) = 3x \) and \( g'(x) = 2x^2 \), then

\( \lim_{x \to 1} \frac{(f(x) + g(x)) - (f(1) + g(1))}{x - 1} \)
(A) 5
(B) 0
(C) 20
(D) None of the above
Answer: (A). The derivative of a sum is the sum of derivatives: \( f'(1) + g'(1) = 3 + 2 = 5 \).

Question 8

Below is the graph of function \( f \) with a maximum at point B:

If \( x_A, x_B, x_C \) are the x-coordinates of points A, B, and C respectively and \( f' \) is the derivative, then:
(A) \( f'(x_A) > 0, f'(x_B) > 0, f'(x_C) > 0 \)
(B) \( f'(x_A) > 0, f'(x_B) = 0, f'(x_C) > 0 \)
(C) \( f'(x_A) > 0, f'(x_B) = 0, f'(x_C) < 0 \)
(D) \( f'(x_A) < 0, f'(x_B) = 0, f'(x_C) > 0 \)
Answer: (C). \( f \) is increasing at A (\( f'(x) > 0 \)), has a maximum at B (\( f'(x) = 0 \)), and decreases at C (\( f'(x) < 0 \)).

References and Links

Calculus questions with answers and Calculus tutorials and problems.