Q: For $f(x) = x^2 \sin(3x) + \sec(x)$, what is $f'(x)$?

$2x\sin(3x) + 3x^2\cos(3x) + \sec(x)\tan(x)$

Question 1

Given the function $f(x) = 2x^2 - 3x + 1$, what is the average rate of change of $f$ from $x = 2$ to $x = 2.6$?

Accepted Answer

$5.2$

Question 2

For the function $f(x) = \frac{1}{x^5}$, use the limit definition to find $f'(x)$. Which of the following is correct?

Accepted Answer

$-5x^{-6}$

Question 3

If $f(x) = \sin(2x) + (x^4 + 1)^{12}$, what is $f'(x)$?

Accepted Answer

$2\cos(2x) + 48x^3(x^4 + 1)^{11}$

Question 4

A bee moves between two locations flying on a straight path. If the function $f(t)$ gives the position of the bee in feet after $t$ seconds of flight, then what are the units of the number $f'(t)$?

Accepted Answer

feet per second

Question 5

For $f(x) = 	an^{-1}(3x)$, find $f'(x)$.

Accepted Answer

$\frac{3}{1 + 9x^2}$

Question 6

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

Accepted Answer

$f(x) = |x|$

Question 7

If $f(x) = \ln(x^2 + 5)$, what is $f'(x)$?

Accepted Answer

$\frac{2x}{x^2 + 5}$

Question 8

For $f(x) = 8x^{-2}$, what is $f'(x)$?

Accepted Answer

$-16x^{-3}$

Question 9

Given $f(x) = e^{5x}$, what is $f'(x)$?

Accepted Answer

$5e^{5x}$

Question 10

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

Accepted Answer

$\frac{1}{x}$

Question 11

For $f(x) = x^2 \sin(3x) + \sec(x)$, what is $f'(x)$?

Accepted Answer

$2x\sin(3x) + 3x^2\cos(3x) + \sec(x)	an(x)$

Question 12

If $f(x) = \cos(8x)$, what is $f'(x)$?

Accepted Answer

$-8\sin(8x)$

Question 13

Given $f(x) = \frac{x^2 + 5x + 4}{x^2 + 3}$, what is $f'(x)$?

Accepted Answer

$\frac{(2x + 5)(x^2 + 3) - (x^2 + 5x + 4)(2x)}{(x^2 + 3)^2}$

Question 14

If $f(x) = \sqrt{x^2 + 4}$, what is $f'(x)$?

Accepted Answer

$\frac{x}{\sqrt{x^2 + 4}}$

Question 15

For $f(x) = \frac{\sin(2x + 1)}{x^2 + 3}$, which of the following derivative rules would you NOT be required to apply?

Accepted Answer

Product Rule

Question 16

If $h(x) = f(x)g(x)$ and $f(3) = 2$, $g(3) = 5$, $f'(3) = 4$, $g'(3) = 6$, what is $h'(3)$?

Accepted Answer

$22$

Question 17

If $h(x) = \frac{f(x)}{g(x)}$ and $f(3) = 2$, $g(3) = 5$, $f'(3) = 4$, $g'(3) = 6$, what is $h'(3)$?

Accepted Answer

$\frac{4 \cdot 5 - 2 \cdot 6}{25}$

Question 18

If $h(x) = f(g(x))$ and $f(3) = 2$, $g(3) = 5$, $f'(3) = 4$, $g'(3) = 6$, what is $h'(3)$?

Accepted Answer

$f'(g(3)) \cdot g'(3)$

Question 19

If $h(x) = \sqrt{f(x)}$ and $f(3) = 2$, $f'(3) = 4$, what is $h'(3)$?

Accepted Answer

$\frac{4}{2\sqrt{2}}$

Calculus Study Guide: Derivatives, Differentiability, and Applications

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