Computation and Properties of the Derivative in Calculus

Questions on the computation and properties of the derivative of a function in calculus are presented. These questions have been designed to help you gain deep understanding of the properties of derivatives . Answers to the questions are also presented.

 Question 1: True or False. If a function is continuous at x = a, then it has a tangent line at x = a. Answer : False. Function f(x) = | x |, for example, is continuous at x = 0 but has no tangent line at x = 0. Question 2: True or False. The derivative of a function at a given point gives the slope of the tangent line at that point. Answer : True. From the definition of the derivative. Question 3: True or False. If f ' is the derivative of f, then the derivative of the inverse of f is the inverse of f '. Answer : False. If g(x) is the inverse of f(x) then its derivative g '(x) is given by. g '(x) = 1 / f ' (g(x)). Question 4: True or False. The derivative of ln a x, where a is a constant, is equal to 1 / x. Answer : True. Question 5: True or False. Rolle's theorem is a special case of the mean value theorem. Answer: True. Question 6: If f(x) = x 3 - 3x 2 + x and g is the inverse of f, then g '(3) is equal to (A) 10 (B) 1 / 10 (C) 1 (D) None of the above Answer : (B). Use g '(x) = 1 / f ' (g(x)) given as the answer to question 3 above to write g '(3) = 1 / f ' (g(3)). First find g(3) which is the solution to the equation f(x) = 3 by definition of the inverse function. x 3 - 3x 2 + x = 3 The above equation has one real solution x = 3. So g(3) = 3, the solution of the above equation. Then compute f '(x) = 3 x 2 - 6 x + 1. f ' (g(3)) = 3 (3) 2 -6 (3) + 1 = 10; and then substitute in the formula that gives g '(3) = 1 / 10. Question 7: True or False. The derivative of f(x) = a x, where a is a constant, is x a x-1. Answer: False. Let y = a x so that ln y = x ln a Differentiate both sides of ln y = x ln a with respect to x to obtain (1 / y) dy / dx = ln a Solve for dy / dx dy / dx = y ln a = a x ln a More references on calculus questions with answers and tutorials and problems .