Question 1

Given $$f(x) = \frac{\cos(x)}{x^2$}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{-2x\cos(x) - x^2$\sin(x)}{x^4$}$$

Question 2

If $$y = 3x$$^{5/2}$$ + 3.5x$$^{1/2}$$ + \ln(x^5$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{15}{2}x$$^{3/2}$$ + \frac{3.5}{2}x$$^{-1/2}$$ + \frac{5x^4$}{x^5$ + 1}$$

Question 3

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

Accepted Answer

$$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$$

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Suppose $$f(x) = x^3$ \sin(x). $$What is $$f'(x)$$?

Accepted Answer

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

Question 7

Given $$y = \ln(x^2$ + 1) + x^3$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 8

If $$f(x) = \frac{\sin(x)}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{x\cos(x) - \sin(x)}{x^2$}$$

Question 9

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{e$$^{-kt}$$}{t}$$, where $$k \u003e 0. $$What is $$\frac{dC}{dt}$$?

Accepted Answer

$$\frac{-ke$$^{-kt}$$t - e$$^{-kt}$$}{t^2$}$$

Question 10

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

Accepted Answer

$$2x\ln(x) + x$$

Question 11

Given $$f(x) = \frac{1}{x^2$ + 1}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 12

If $$y = \ln(x^3$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

A population grows according to $$P(t) = t^2$ e$$^{2t}$$. What is $$\frac{dP}{dt}$$?

Accepted Answer

$$2t e$$^{2t}$$ + 2t^2$ e$$^{2t}$$

Question 14

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

Accepted Answer

$$2x\ln(x) + x$$

Question 15

Given $$y = \ln(x^2$ + 5)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 16

If $$f(x) = \frac{\cos(x)}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{-x\sin(x) - \cos(x)}{x^2$}$$

Question 17

A chemical reaction's rate is given by $$R(t) = \frac{\ln(t)}{t^2$}. $$What is $$\frac{dR}{dt}$$?

Accepted Answer

$$\frac{1 - 2\ln(t)}{t^3$}$$

Question 18

If $$y = \ln(x^2$ + 1) + 3x$$^{1/2}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1} + \frac{3}{2}x$$^{-1/2}$$

Question 19

Given $$f(x) = \frac{\sin(x)}{x^2$}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{x^2$\cos(x) - 2x\sin(x)}{x^4$}$$

Calculus Study Notes: Derivatives of Quotients and Logarithmic Functions

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