Q: Given $$\ln x \tan y = y^2$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

$$\frac{\tan y / x}{2y + \ln x \sec^2$ y}$$

Question 1

Given the implicit equation $$4x^2 + 24x$$ + $$9y^2 + 36y = 36$, what is the correct expression for $$\frac{dy}{dx}$$ after differentiating both sides with respect to x$ and solving for $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{8x - 24}{18y + 36}$$

Question 2

For the curve defined by $$4x^2 + 24x$$ + $$9y^2 + 36y = 36$, at which points does the curve have horizontal tangents?

Accepted Answer

At $$(3, 0)$$ and $$(3, 4)$$

Question 3

Suppose $$y is $$defined implicitly by $$x^2$$ + $$y^2 = 25$. What is $$\frac{$$d^2y$$}{$$dx^2$$}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{-x^2$ - y^2$}{y^3$}$$

Question 4

A square metal plate is expanding such that each side grows at a constant rate of $$2 cm/$$min. When the side length is $$7 cm$$, what is the rate of change of the area?

Accepted Answer

$$28 cm$^2/$$min

Question 5

Which of the following best describes the chain rule as applied in implicit differentiation?

Accepted Answer

Multiply the derivative of $$y-$$terms by $$\frac{dy}{dx}$$

Question 6

Given $$y = x^x$$, use logarithmic differentiation to find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 7

For the implicit equation $$x^2 + xy$$ + $$y^2 = 7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{-2x - y}{x + 2y}$$

Question 8

A spherical balloon is being inflated so that its radius increases at a rate of $$3 cm/$$min. What is the rate of change of its volume when the radius is $$10 cm$$? (Recall $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$1200\pi cm$^3/$$min

Question 9

Which of the following is a correct application of logarithmic differentiation for $$y = \frac{(x^2$ + 1)^5$}{\sqrt{x}}$$?

Accepted Answer

$$\ln y = 5\ln(x^2$ + 1) - \frac{1}{2}\ln x$$

Question 10

Given $$\ln x 	an y = y^2$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{	an y / x}{2y + \ln x \sec^2$ y}$$

Question 11

If $$y is $$defined implicitly by $$e^{xy} = x + y$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{e$$^{xy}$$y - 1}{e$$^{xy}$$x - 1}$$

Question 12

A ladder $$10 ft $$long is leaning against a wall. If the bottom slides away from the wall at $$2 ft/s$$, how fast is the top sliding down when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.2 ft/s$$

Question 13

Which of the following is NOT a correct step in implicit differentiation?

Accepted Answer

Writing $$\frac{dy}{dx} on $$both sides instead of $$\frac{d}{dx}$$

Question 14

Given $$A = s^2$ for a square, and $$\frac{ds}{dt} = k$$ (constant), what is the general formula for $$\frac{dA}{dt}$$?

Accepted Answer

$$2s \frac{ds}{dt}$$

Question 15

For the ellipse $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{-4x}{9y}$$

Question 16

If $$y is $$defined implicitly by $$x^3$$ + $$y^3 = 6xy$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 17

Which of the following is a common mistake when performing implicit differentiation?

Accepted Answer

Writing $$\frac{dy}{dx} on $$both sides instead of $$\frac{d}{dx}$$

Question 18

If $$y = (x^2$ + 1$$)^{x}$$, what is $$\frac{dy}{dx}$$ using logarithmic differentiation?

Accepted Answer

$$(x^2 + 1)^x$$ \left( \$$ln(x^2 + 1$$) + \frac{2x}{$$x^2 + 1$$} \right)$$

Implicit Differentiation, Tangents, Logarithmic Differentiation, and Related Rates

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