Question 1

Given $$y = 	an(x)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\sec^2$(x)$$

Question 2

Find the derivative of $$y = e$$^{-3x}$$ with respect to x$.

Accepted Answer

$$-3e^{-3x}$$

Question 3

If $$y = (x^2$ - 2x + 2$$)^{\frac{x}$${2}}$$, which of the following is the correct expression for $$\dfrac{dy}{dx}$$?

Accepted Answer

$$(x^2 - 2x$$ + $$2)^{\frac{x}$${2}} \left[ \$$ln(x^2 - 2x + 2$$) \cdot \frac{d}{dx}\left(\frac{x}{2}\right) + \frac{x}{2} \cdot \frac{2x - 2}{$$x^2 - 2x + 2$$} \right]$$

Question 4

Find $$\dfrac{dy}{dx}$$ for $$y = \left( \frac{x-1}{x+1} \right)$$ using implicit differentiation.

Accepted Answer

$$\dfrac{2}{(x+1)^2$}$$

Question 5

Given $$y = \ln\left(\frac{x}{2}\right)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{1}{x}$$

Question 6

Find the equation of the tangent line to $$y = 2 \arccos\left( \frac{x}{\sqrt{2}} \right) at$$ $$x = 1.$$

Accepted Answer

$$y = \pi x + 2 - \pi$$

Question 7

If $$\$$theta^{1/2}$$ + r$$^{1/2}$$ = 1$$, find $$\dfrac{dr}{d	heta}$$ using implicit differentiation.

Accepted Answer

$$-\dfrac{r$$^{1/2}$$}{\$$theta^{1/2}$$}$$

Question 8

Given $$y = e$$^{x \ln(\sin x)}$$, find $$\dfrac{dy}{dx}$$.

Accepted Answer

$$e^{x \ln(\sin x)}$$ \left[ \ln(\sin x) + x \cot x \right]$$

Question 9

If $$y = (\sin x)^x$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$(\sin x)^x \left[ \ln(\sin x) + x \cot x \right]$$

Question 10

A biologist models the growth of a bacteria population by $$y = e$$^{-3x}$$, where x$ is time in hours. What is the instantaneous rate of change of the population at x = 2$ hours?

Accepted Answer

$$-3e^{-6}$$

Question 11

A physicist uses $$y = x^\pi to $$model a process. What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\pi x$$^{\pi - 1}$$

Question 12

Find the derivative of $$y = \ln\left(\frac{x}{2}\right)$$ and interpret its meaning in the context of economics, where $$x is $$the price of a product.

Accepted Answer

The derivative is $$\dfrac{1}{x}$$, representing the rate of change of the logarithm of price with respect to price.

Question 13

A chemist is studying a reaction where the concentration $$C$$ changes according to $$C = (x^2$ - 2x + 2$$)^{x/2}$$. What is the best method to find $$\dfrac{dC}{dx}$$?

Accepted Answer

Use logarithmic differentiation.

Question 14

Given $$y = 	an(x)$$, which of the following is true about its derivative?

Accepted Answer

It is always positive for $$x in$$ $$(0, \frac{\pi}{2}).$$

Question 15

If $$y = e$$^{-3x}$$ models the decay of a radioactive substance, what does the sign of $$\dfrac{dy}{dx}$$ indicate?

Accepted Answer

The substance is decreasing over time.

Question 16

A medical researcher models the spread of a disease by $$y = (\sin x)^x. $$What is the best way to differentiate this function?

Accepted Answer

Take the natural logarithm of both sides and use implicit differentiation.

Question 17

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

Accepted Answer

$$\pi x$$^{\pi - 1}$$

Question 18

If $$\$$theta^{1/2}$$ + r$$^{1/2}$$ = 1$$, and $$	heta$$ increases, what happens to $$r$$?

Accepted Answer

It decreases.

Question 19

Find $$\dfrac{dy}{dx}$$ for $$y = e$$^{x \ln(\sin x)}$$ and explain the role of the chain rule in your solution.

Accepted Answer

The chain rule is used to differentiate the exponent $$x \ln(\sin x)$$, resulting in $$e^{x \ln(\sin x)}$$ \left[ \ln(\sin x) + x \cot x \right]$$.

Calculus Study Notes: Differentiation, Tangent Lines, and Implicit Differentiation

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