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.
Questions with SolutionsQuestion 1True 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 2True 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 3True 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.
Question 4True or False. The derivative of ln a x, where a is a constant, is equal to 1 / x.Answer : True.
Question 5True or False. Rolle's theorem is a special case of the mean value theorem.Answer: True.
Question 6If 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 7True or False. The derivative of f(x) = a^{ x}, where a is a constant, is x a ^{ x1}.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 .
