Question 1

Given $$y = x^x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$x^x (1 + \ln(x))$$

Question 2

If $$y = \log_2$ x$$, what is $$\frac{d^3$y}{dx^3$}$$?

Accepted Answer

$$\frac{2}{x^3$ (\ln 2)}$$

Question 3

A body moves along the x-axis with position $$s = \arctan(2t) - t. At $$what time $$t \u003e 0$$ does the body change direction?

Accepted Answer

$$t = 1$$

Question 4

Given $$y = \sin$$^{-1}$$(e^{-x}$$) + \$$sec^{-1}(e^x$$)$$ for x$$ \geq 0$$, simplify y$.

Accepted Answer

$$y = \frac{\pi}{2}$$

Question 5

Find the equation of the tangent line to $$y = x^4$ + \ln x at$$ $$x = e.$$

Accepted Answer

$$y = (4e^3$ + \frac{1}{e})(x-e) + e^4$ + 1$$

Question 6

Use the definition of the derivative as a limit to find the derivative of $$f(x) = \frac{1}{x}.$$

Accepted Answer

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

Question 7

Suppose $$f(x) is a $$differentiable function with an inverse and the graph of $$f$$ passes through $$(1,3)$$ with slope $$1/2$$ there. What is the slope of the tangent to $$f^{-1} at$$ $$x=3$$?

Accepted Answer

$$2$$

Question 8

Find the equation of the tangent line to $$y = \sin(\arccos(2x)) at$$ $$x = 1/4.$$

Accepted Answer

$$y = \frac{\sqrt{3}}{2} - \frac{2}{\sqrt{3}}(x - 1/4)$$

Question 9

The volume of an expanding sphere is increasing at a rate of $$16\pi ft$^3/h $$when $$r = 1 ft. $$How fast is the radius increasing at this instant?

Accepted Answer

$$4 ft/h$$

Question 10

Given $$y = e$$^{x^2}$$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$2x e$$^{x^2}$$

Question 11

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

Accepted Answer

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

Question 12

A medication is administered such that its concentration in the bloodstream at time $$t (in $$hours) is $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

$$-1e^{-0.6} mg/L $$per hour

Question 13

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 14

A bacteria culture grows so that its population at time $$t (in $$hours) is $$P(t) = 100e$$^{0.05t}$$. How fast is the population growing at t = 10$ hours?

Accepted Answer

$$5e^{0.5}$$ individuals/hour

Question 15

Given $$y = x^2$ \sin x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 16

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

Accepted Answer

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

Question 17

A car's position is given by $$s(t) = t^3$ - 6t^2$ + 9t (in $$meters, $$t in $$seconds). At what time does the car come to rest?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 18

If $$y = \sqrt{x^2$ + 9}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 9}}$$

Question 19

Given $$y = \ln(\sin x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cot x$$

Question 20

If $$y = x^3$ e^x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$e^x(x^3$ + 3x^2$)$$

Calculus Study Notes: Derivatives, Tangents, and Applications

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