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 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:__

lim [e^{ 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

f '(a) = lim [f(x) - f(a)] / (x - a) as x approaches a.
The given limit is the derivative of e

^{ x} at x = 0.

__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

lim [f(x) - f(4)] / (x - 4) as x approaches 4.

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

f '(x) = 3 x and g '(x) = 2 x^{ 2}

then the limit

lim [(f(x) + g(x)) - (f(1) + g(1))] / (x - 1) as x approaches 1.

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.
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