# 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) . 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

^{ x}-1] / x as x approaches 0

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 2 g '(x) . g(x)].

### Question 5

**True or False**. The derivative of f(x) . g(x) is equal to f '(x) g(x) + f(x).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 is equal to f '(4).

### Question 7

If f(x) and g(x) are differentiable functions such that^{ 2}

then 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 xA, xB and xC are the x coordinates of points A, B and C respectively and f ' is the first derivative of f, then

(A) f '(xA) > 0 , f '(xB) > 0 and f '(xC) > 0

(B) f '(xA) > 0 , f '(xB) = 0 and f '(xC) > 0

(C) f '(xA) > 0 , f '(xB) = 0 and f '(xC) < 0

(D) f '(xA) < 0 , f '(xB) = 0 and f '(xC) > 0

__Answer :__(C). f is increasing (f '(x) > 0) at point A, decreasing (f '(x) < 0) at C and has a maximum (f '(x) = 0) at B.

More references on calculus questions with answers and tutorials and problems .