# Questions and Answers on Derivatives in Calculus

A set of questions on the concepts of the derivative of a function in calculus are presented with their answers. These questions have been designed to help you **gain deep understanding of the concept of derivatives** which is of major importance in calculus.

## Questions with Solutions

### Question 1

If functions \( f \) and \( g \) are such thatwhere \( k \) 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 sum of two functions is equal to the sum of the derivatives of the two functions and also the derivative of constant is equal to zero.

### 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).

The derivative of the composition of two functions is given by the chain rule.

### Question 3

is equal to

(A) 1

(B) 0

(C) is of the form 0 / 0 and cannot be calculated.

__Answer :__(A).

The definition of the derivative at \( x = a \) is given by

For \( f(x) = e^x \), \( f'(x) = e^x \)

The given limit is the derivative of \( e^x \) at \( x = 0 \) which is \( e^0 = 1 \)

### Question 4

**True or False**. The derivative of \( [g(x)]^2 \) is equal to \( [g'(x)]^2 \).

__Answer :__False.

The derivative of \( [g(x)]^2 \) is equal to \( 2g'(x) \cdot g(x) \).

### Question 5

**True or False**. The derivative of \( f(x) \cdot g(x) \) is equal to \( f'(x) \cdot g(x) + f(x) \cdot g'(x) \).

__Answer:__True.

### Question 6

If \( f(x) \) is a differentiable function such that \( f'(0) = 2 \), \( f'(2) = -3 \) and \( f'(5) = 7 \) then the limit

is equal to

(A) 2

(B) -3

(C) 7

(D) None of the above

__Answer :__(D).

The given limit \( \lim_{{x \to 4}} \dfrac{{f(x) - f(4)}}{{x - 4}} \) is equal to \( f'(4) \) by definition of the derivative.

### Question 7

If \( f(x) \) and \( g(x) \) are differentiable functions such thatthen the limit

is equal to

(A) 5

(B) 0

(C) 20

(D) None of the above

__Answer :__(A).

The given limit is the definition of the derivative of \( f(x) + g(x) \) at \( x = 1 \). The derivative of the sum is equal to the sum of the derivatives. Hence the given limit is equal to \( f'(1) + g'(1) = 5 \).

### Question 8

Below is the graph of function \( f \). This graph has a maximum point at B.If \( x_A \), \( x_B \) and \( x_C \) are the \( x \) coordinates of points A, B and C respectively and \( f' \) is the first derivative of \( f \), then

(A) \( f'(x_A) > 0 \) , \( f'(x_B) > 0 \) and \( f'(x_C) > 0 \)

(B) \( f'(x_A) > 0 \) , \( f'(x_B) = 0 \) and \( f'(x_C) > 0 \)

(C) \( f'(x_A) > 0 \) , \( f'(x_B) = 0 \) and \( f'(x_C) < 0 \)

(D) \( f'(x_A) \lt 0 \) , \( f'(x_B) = 0 \) and \( f'(x_C) > 0 \)

__Answer :__(C).

\( f \) is increasing (\( f'(x) > 0 \)) at point A, decreasing (\( f'(x) \lt 0 \)) at C and has a maximum (\( f'(x) = 0 \)) at B.