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

Question 1

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

Accepted Answer

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

Question 2

Find the equation of the tangent line to the circle $$x^2$$ + $$y^2 = 25$ at the point (3$$,4)$$.

Accepted Answer

$$y - 4 = -\frac{3}{4}(x - 3)$$

Question 3

For the curve defined by $$x^3$$ + $$y^3 = 6xy$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

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

Question 4

At what point in the first quadrant does the curve $$x^3$$ + $$y^3 = 6xy$ have a horizontal tangent?

Accepted Answer

$$(2^{2/3}$$, $$2^{4/3}$$)$$

Question 5

If $$\sin(x + y) = y^2$ \cos x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{-(y^2$ \sin x + \cos(x + y))}{\cos(x + y) - 2y \cos x}$$

Question 6

Find $$\frac{d}{dx}(\sin$$^{-1}$$ x)$$ for $$|x| \u003c 1.$$

Accepted Answer

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

Question 7

Find $$\frac{d}{dx}(\cos$$^{-1}$$ x)$$ for $$|x| \u003c 1.$$

Accepted Answer

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

Question 8

Find $$\frac{d}{dx}(	an$$^{-1}$$ x)$$ for all real $$x.$$

Accepted Answer

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

Question 9

Given $$x^4$$ + $$y^4 = 16$, find $$\frac{$$d^2y$$}{$$dx^2$$}$$ in terms of x$ and y$.

Accepted Answer

$$-\frac{3x^2$(x^4$$ + $$y^4$$)}{$$y^7$$}$

Question 10

If $$y = (\sin$$^{-1}$$ x$$)^{-1}$$, what is $$\frac{dy}{dx}$$ for $$|x| \u003c 1$$?

Accepted Answer

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

Question 11

Let $$f(x) = x 	an$$^{-1}$$ \sqrt{x}. $$What is $$f'(x)$$ for $$x \u003e 0$$?

Accepted Answer

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

Question 12

Given $$x^2 + xy$$ - $$y^2 = 4$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Suppose $$e^y \sin x = x + x y. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1 + y - e^y \cos x}{e^y \sin x - x}$$

Question 15

Find $$\frac{dy}{dx} if$$ $$	an$$^{-1}$$(x^2 y) = x + x$$ $$y^2$.

Accepted Answer

$$\frac{(1 + x^4$ y^2$)(1 + y^2$) - 2x y}{x^2$ - 2x y (1 + x^4$ y^2$)}$$

Question 16

Find the equation of the tangent line to $$y \sin(2x) = x \cos(2y) at$$ $$\left(\frac{\pi}{2}, \frac{\pi}{4}\right).$$

Accepted Answer

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

Question 17

Given $$x^2 + x y$$ + $$y^2 = 3$ at the point (1$$,1)$$, what is the equation of the tangent line?

Accepted Answer

$$y = -x + 2$$

Question 18

Which of the following is the correct derivative of $$\sec$$^{-1}$$ x$$ for $$|x| \u003e 1$$?

Accepted Answer

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

Question 19

Suppose $$x^4(x + y$$) = $$y^2(3x - y$$)$$. What is the denominator of $$\frac{dy}{dx}$$ after collecting like terms?

Accepted Answer

$$x^4 - 6x y$$ + $$3y^2$$

Implicit Differentiation and Derivatives of Inverse Trigonometric Functions

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