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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