Problem Set: Implicit Differentiation

For the following exercises (1-10), use implicit differentiation to find $\frac{dy}{dx}$ .

1. $x^2-y^2=4$

2. $6x^2+3y^2=12$

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$\frac{dy}{dx}=\frac{-2x}{y}$

3. $x^2 y=y-7$

4. $3x^3+9xy^2=5x^3$

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

5. $xy- \cos (xy)=1$

6. $y\sqrt{x+4}=xy+8$

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$\frac{dy}{dx}=\large \frac{y-\frac{y}{2\sqrt{x+4}}}{\sqrt{x+4}-x}$

7. $−xy-2=\dfrac{x}{7}$

8. $y \sin(xy)=y^2+2$

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$\frac{dy}{dx}=\large \frac{y^2 \cos(xy)}{2y- \sin(xy)-xy \cos xy}$

9. $(xy)^2+3x=y^2$

10. $x^3 y+xy^3=-8$

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\frac{d y}{d x} = \frac{- 3 x^{2} y - y^{3}}{x^{3} + 3 x y^{2}}

$\frac{dy}{dx}=\large \frac{-3x^2 y-y^3}{x^3+3xy^2}$

For the following exercises (11-16), find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

11. [T] $x^4 y-xy^3=-2, \,\,\, (-1,-1)$

12. [T] $x^2 y^2+5xy=14, \,\,\, (2,1)$

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The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope −1/2 and y intercept 2.

y = - \frac{1}{2} x + 2

$y=-\frac{1}{2}x+2$

13. [T] $\tan (xy)=y, \,\,\, (\frac{\pi}{4},1)$

14. [T] $xy^2 + \sin(\pi y)-2x^2=10, \,\,\, (2,-3)$

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The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1/(π + 12) and y intercept −(3π + 38)/(π + 12).

y = \frac{1}{π + 12} x - \frac{3 π + 38}{π + 12}

$y=\large \frac{1}{\pi +12}x-\frac{3\pi +38}{\pi +12}$

15. [T] $\frac{x}{y}+5x-7=-\frac{3}{4}y, \,\,\, (1,2)$

16. [T] $xy+ \sin (x)=1, \,\,\, (\frac{\pi}{2},0)$

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The graph starts in the third quadrant near (−5, 0), remains near 0 until x = −4, at which point it decreases until it reaches near (0, −5). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0.

y = 0

$y=0$

17. [T] The graph of a folium of Descartes with equation $2x^3+2y^3-9xy=0$ is given in the following graph.

A folium is graphed which has equation 2x3 + 2y3 – 9xy = 0. It crosses over itself at (0, 0).

Find the equation of the tangent line at the point $(2,1)$ . Graph the tangent line along with the folium.
Find the equation of the normal line to the tangent line in a. at the point $(2,1)$ .

18. For the equation $x^2+2xy-3y^2=0$ ,

Find the equation of the normal to the tangent line at the point $(1,1)$ .
At what other point does the normal line in a. intersect the graph of the equation?

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a. $y=−x+2$
b. $(3,-1)$

19. Find all points on the graph of $y^3-27y=x^2-90$ at which the tangent line is vertical.

20. For the equation $x^2+xy+y^2=7$ ,

Find the $x$ -intercept(s).
Find the slope of the tangent line(s) at the $x$ -intercept(s).
What does the value(s) in b. indicate about the tangent line(s)?

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a. $(\pm \sqrt{7},0)$
b. Slope is -2 at both intercepts
c. They are parallel since the slope is the same at both intercepts.

21. Find the equation of the tangent line to the graph of the equation $\sin^{-1} x+\sin^{-1} y=\frac{\pi}{6}$ at the point $(0,\frac{1}{2})$ .

22. Find the equation of the tangent line to the graph of the equation $\tan^{-1}(x+y)=x^2+\frac{\pi}{4}$ at the point $(0,1)$ .

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$y=−x+1$

23. Find $y^{\prime}$ and $y''$ for $x^2+6xy-2y^2=3$ .

24. [T] The number of cell phones produced when $x$ dollars is spent on labor and $y$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $60x^{\frac{3}{4}}y^{\frac{1}{4}}=3240$ .

Find $\frac{dy}{dx}$ and evaluate at the point $(81,16)$ .
Interpret the result of a.

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25. [T] The number of cars produced when $x$ dollars is spent on labor and $y$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $30x^{\frac{1}{3}}y^{\frac{2}{3}}=360$ .

(Both $x$ and $y$ are measured in thousands of dollars.)

Find $\frac{dy}{dx}$ and evaluate at the point $(27,8)$ .
Interpret the result of a.

26. The volume of a right circular cone of radius $x$ and height $y$ is given by $V=\frac{1}{3}\pi x^2 y$ . Suppose that the volume of the cone is $85\pi \, \text{cm}^3$ . Find $\frac{dy}{dx}$ when $x=4$ and $y=16$ .

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$\frac{dy}{dx}=-8$

For the following exercises (27-28), consider a closed rectangular box with a square base with side $x$ and height $y$ .

27. Find an equation for the surface area of the rectangular box, $S(x,y)$ .

28. If the surface area of the rectangular box is 78 square feet, find $\frac{dy}{dx}$ when $x=3$ feet and $y=5$ feet.

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$\frac{dy}{dx}=-2.67$

For the following exercises (29-31), use implicit differentiation to determine $y^{\prime}$ . Does the answer agree with the formulas we have previously determined?

29. $x= \sin y$

30. $x= \cos y$

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$y^{\prime}=-\frac{1}{\sqrt{1-x^2}}$

31. $x= \tan y$

Derivatives: Supplemental Content

Problem Set: Implicit Differentiation

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