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.
Question 1:
If functions f and g are such that
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 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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