Question 1

Which of the following best describes the geometric interpretation of the derivative $$f'(x) at a $$point $$x$$?

Accepted Answer

It is the slope of the tangent line to the graph of $$f at $$the point $$(x, f(x)).$$

Question 2

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

Accepted Answer

$$6x$$

Question 3

If $$f(x) = (5x^3$ + 2)(\sqrt{x} + 1)$$, which rule should be used to find $$f'(x)$$?

Accepted Answer

Product Rule

Question 4

Find the derivative of $$y = e$$^{2x}$$ with respect to x$.

Accepted Answer

$$2e^{2x}$$

Question 5

Which of the following is the correct derivative of $$y = \ln(3x^2$ + 1)$$?

Accepted Answer

$$\frac{6x}{3x^2$ + 1}$$

Question 6

If $$y = \sin(3x^2$) + \cos(4x^3$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$6x\cos(3x^2$) - 12x^2$\sin(4x^3$)$$

Question 7

Given $$y = x^3$ + 3x^2$ - 7$$, for which interval is $$f(x)$$ increasing?

Accepted Answer

$$x \u003e -1 or$$ $$x \u003c -2$$

Question 8

Which of the following is the correct derivative of $$y = 	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 9

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 10

Given the parametric equations $$x = t^2$, y$$ = $$2t^2 + 1$, find $$\frac{dy}{dx}$$ in terms of t$.

Accepted Answer

$$\frac{4t}{2t}$$

Question 11

Which of the following is the correct equation for the tangent to the curve $$y = f(x) at $$the point $$(x_1$, y_1$)$$?

Accepted Answer

$$y - y_1$ = f'(x_1$)(x - x_1$)$$

Question 12

For the function $$f(x) = x^4$ + x^2$ - 5x + 5$$, in which interval is the function concave upwards?

Accepted Answer

Where $$f''(x) \u003e 0$$

Question 13

Which of the following is NOT a possible reason for a point to be a critical point of $$f(x)$$?

Accepted Answer

$$f(x) is $$undefined

Question 14

If $$y = \ln(3x - 2)^6$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{18}{3x - 2}$$

Question 15

Given $$y = \frac{2x^2$ - 3}{2x + 3}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{4x(2x + 3) - 2(2x^2$ - 3)}{(2x + 3)^2$}$$

Question 16

Which of the following statements about the second derivative test is correct?

Accepted Answer

If $$f''(c) = 0$$, the test is inconclusive.

Question 17

If $$y = \ln\left(\frac{x-1}{x+1}\right)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 18

A curve is given by $$x = \sin^2$ t$$, $$y = \cos t. $$What is the slope of the tangent to the curve at $$t = \frac{\pi}{3}$$?

Accepted Answer

$$-\frac{1}{\sqrt{3}}$$

Question 19

Which of the following is the correct derivative of $$y = \log_2$(3x^2$$)$?

Accepted Answer

$$\frac{6x}{3x^2$ \ln 2}$$

Chapter 3: Differentiation – Rules, Techniques, and Applications

Study Guide - Practice Questions