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 Solutions

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