Question 1

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

Accepted Answer

$$f'(x) = -\frac{\sin(x)}{x^2$} - \frac{2\cos(x)}{x^3$}$$

Question 2

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

Accepted Answer

$$\frac{15}{2}x$$^{3/2}$$ + \frac{3}{5}x$$^{-4/5}$$ + \frac{5x^4$}{x^5$ + 1} + 15x^2$ + nx$$^{n-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

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

Accepted Answer

$$C'(t) = -\frac{ke$$^{-kt}$$}{t} - \frac{e$$^{-kt}$$}{t^2$}$$

Question 7

Suppose $$f(x) = x^2$ \sin(x). $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x\sin(x) + x^2$\cos(x)$$

Question 8

Given $$y = \ln(x^2$ + 1) + 3x^5$, what is $$\frac{dy}{dx}$$ evaluated at x = 1$?

Accepted Answer

$$\frac{2}{2} + 3 \cdot 5 \cdot 1^4$ = 1 + 15 = 16$$

Question 9

If $$y = \ln(x^2$ + 1)$$, what is the slope of the tangent line at $$x = 0$$?

Accepted Answer

$$0$$

Question 10

A population grows according to $$P(t) = 5e$$^{0.03t}$$. What is the instantaneous rate of change at t = 10$?

Accepted Answer

$$P'(10) = 0.15e$$^{0.3}$$

Question 11

Given $$f(x) = \frac{2x\cos^2$(x) + (x^2$ + 1)\sin(x)\cos(x)}{\cos^2$(x)}$$, which of the following simplifies $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x\cos(x) + (x^2$ + 1)\sin(x)$$

Question 12

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

Accepted Answer

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

Question 13

A chemical reaction's rate is given by $$R(t) = \ln(t^2$ + 1). $$What is the rate of change of $$R at$$ $$t = 2$$?

Accepted Answer

$$\frac{4}{5}$$

Question 14

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

Accepted Answer

$$x = 1$$

Question 15

Given $$y = x^n$$, which of the following is the correct derivative?

Accepted Answer

$$y' = nx$$^{n-1}$$

Question 16

Suppose $$y = \ln(x^2$ + 1) + 3x^5$. What is the value of $$\frac{dy}{dx}$$ at x = 0$?

Accepted Answer

$$0$$

Question 17

If $$y = \ln(x^2$ + 1)$$, what is the value of $$\frac{dy}{dx} as$$ $$x 	o \infty$$?

Accepted Answer

$$0$$

Question 18

Given $$y = \ln(x^2$ + 1)$$, which of the following best describes the concavity of $$y$$ for $$x \u003e 0$$?

Accepted Answer

Concave down for $$x \u003e 1$$, concave up for $$0 \u003c x \u003c 1$$

Calculus Study Notes: Derivatives of Quotients and Logarithmic Functions

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