Q: For $$y = e^x \tan y$$, what is $$\frac{dy}{dx}$$?

$$\frac{dy}{dx} = \frac{e^x \tan y}{1 - e^x \sec^2$ y}$$

Question 1

Which of the following best describes an implicitly defined function?

Accepted Answer

A function where $$y is $$defined by an equation involving both $$x$$ and $$y$$, and $$y is $$not isolated.

Question 2

Given the equation $$x^2$$ + $$y^2 = 25$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{dy}{dx} = \frac{-x}{y}$$

Question 3

For the equation $$y = \frac{x+3}{y-3}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{1}{2y-3}$$

Question 4

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

Accepted Answer

$$\frac{dy}{dx} = \frac{\cos x}{1 - \cos y}$$

Question 5

Given $$e^{xy} = 3y$, find $$\frac{dy}{dx}$$ in terms of x$ and y$.

Accepted Answer

$$\frac{dy}{dx} = \frac{y e$$^{xy}$$}{3 - x e$$^{xy}$$}$$

Question 6

For $$y = e^x 	an y$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{e^x 	an y}{1 - e^x \sec^2$ y}$$

Question 7

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

Accepted Answer

$$\frac{dy}{dx} = \frac{1}{e^x + 2y \sin y^2$}$$

Question 8

A curve is defined by $$4x^2 + xy$$ + $$5y^2 = 52$. What is the slope of the curve at the point (1$$,3)$$?

Accepted Answer

$$\frac{11}{31}$$

Question 9

Which rule is most commonly used when differentiating implicitly defined functions involving products of $$x$$ and $$y$$?

Accepted Answer

Product rule

Question 10

In the context of implicit differentiation, when differentiating $$y^2$$ with respect to $$x$$, what is the result?

Accepted Answer

$$2y \frac{dy}{dx}$$

Question 11

A medical researcher models the spread of a disease with $$x^2$$ + $$y^2$$ = $$r^2$$, where $$x$$ and $$y$$ are spatial coordinates and $$r is $$the radius of infection. If the infection radius $$r is $$increasing at a rate of $$2$$ units per day, what is $$\frac{dy}{dx} at $$the point $$(x, y)$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{-x}{y}$$

Question 12

If $$y is $$defined implicitly by $$y^2 - 3y = x + 3$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{1}{2y - 3}$$

Question 13

A chemical reaction is modeled by $$e^{xy} = 3y$. If x$ represents time and y$ the concentration, what is the rate of change of concentration with respect to time?

Accepted Answer

$$\frac{dy}{dx} = \frac{y e$$^{xy}$$}{3 - x e$$^{xy}$$}$$

Question 14

If $$y = e^x 	an y$$, which differentiation rules must be applied to find $$\frac{dy}{dx}$$?

Accepted Answer

Product rule and chain rule

Question 15

Given $$\sin x + \sin y = y$$, which step is necessary to isolate $$\frac{dy}{dx}$$?

Accepted Answer

Group all terms with $$\frac{dy}{dx} on $$one side and factor.

Question 16

A curve is defined by $$x^2$$ + $$y^2 = 25$. At the point (3$$, 4)$$, what is the slope of the tangent line?

Accepted Answer

$$\frac{-3}{4}$$

Question 17

If $$y is $$defined implicitly by $$y^2 - 3y = x + 3$, and y = 4$ at x = 5$, what is $$\frac{dy}{dx}$$ at this point?

Accepted Answer

$$\frac{1}{2 \cdot 4 - 3} = \frac{1}{5}$$

Implicit Differentiation: 3.8

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