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 SolutionsQuestion 1If 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 2If 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 3is 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 4True 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 5True or False. The derivative of f(x) . g(x) is equal to f '(x) g(x) + f(x).g '(x).Answer: True.
Question 6If 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 7If 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 8Below 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 .
