$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x = a.$$

Question 2

Given $$f(x) = x^2$ + 3x$$, what is $$f'(x)$$?

Accepted Answer

$$2x + 3$$

Question 3

Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

If $$f(x) = 5$$, what is $$f'(x)$$?

Accepted Answer

$$0$$

Question 5

Which rule should be used to differentiate $$f(x) = (2x^3$)(\sin x)$$?

Accepted Answer

Product Rule

Question 6

Find the derivative of $$f(x) = \frac{x^2$ + 1}{x}.$$

Accepted Answer

$$\frac{x^2$ - 1}{x^2$}$$

Question 7

If $$f(x) = x^4$ and g(x$$) = \ln x$$, what is the derivative of h(x) = f(x)g(x$$)$$?

Accepted Answer

$$x^4$$ \cdot \frac{1}{x} + $$4x^3$$ \ln x$$

Question 8

Which of the following statements is true about the derivative as a function?

Accepted Answer

The derivative assigns to each $$x$$ the slope of the tangent line to $$f(x) at$$ $$x.$$

Question 9

A car's position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the car's instantaneous velocity at $$t = 4$$ seconds?

Accepted Answer

$$26$$

Question 10

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{u(x)}{v(x)}$$?

Accepted Answer

$$\frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$$

Question 11

If $$f(x) = x^3$ - 2x$$ and $$g(x) = x^2$ + 1$$, find $$\frac{d}{dx}[f(x)g(x)].$$

Accepted Answer

$$(x^3 - 2x$$) \cdot 2x + $$(x^2 + 1$$) \cdot $$(3x^2 - 2$$)$$

Question 12

Which of the following is NOT a valid interpretation of the derivative?

Accepted Answer

The area under the curve $$f(x)$$ from $$x = a to$$ $$x = b.$$

Question 13

If $$y = 7x^5$ - 4x^2$ + 3$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$35x^4 - 8x$

Question 14

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-2.5e^{-1}$$

Question 15

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 16

If $$f(x) = x^2$ and g(x) = x + 1$, what is $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$$?

Accepted Answer

$$\frac{(x + 1)2x - x^2$ \cdot 1}{(x + 1)^2$}$$

Question 17

Which of the following is the correct derivative of $$f(x) = \sin(x) \cdot e^x$$?

Accepted Answer

$$\sin(x) e^x + \cos(x) e^x$$

Question 18

A population grows according to $$P(t) = 1000e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$30e^{0.3}$$

Question 19

If $$f(x) = x^2$ and g(x$$) = \sin x$$, what is $$\frac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

$$x^2$$ \cos x + 2x \sin x$$

Study Notes: Derivatives and Differentiation in Calculus

Study Guide - Practice Questions