3.11: Implicit Differentiation Exercises

3.8: Implicit Differentiation

For the following exercises, use implicit differentiation to find $\frac{dy}{dx}$.

300) $x^2−y^2=4$

301) $6x^2+3y^2=12$

Answer:

$\frac{dy}{dx}=\frac{−2x}{y}$

Solution: $\frac{dy}{dx}=\frac{−2x}{y}$

302) $x^2y=y−7$

303) $3x^3+9xy^2=5x^3$

Answer:

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

304) $xy−\cos(xy)=1$

305) $y\sqrt{x+4}=xy+8$

Answer:

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

306) $−xy−2=\frac{x}{7}$

307) $y\sin(xy)=y^2+2$

Answer:

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

308) $(xy)^2+3x=y^2$

309) $x^3y+xy^3=−8$

Answer:

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

For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Just for observation, use a calculator or computer software to graph the function and the tangent line.

310) $[T] x^4y−xy^3=−2,(−1,−1)$

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

Answer:

$y=−\frac{1}{2}x+2$

312) $[T] tan(xy)=y,(\frac{π}{4},1)$

313) $[T] xy^2+sin(πy)−2x^2=10,(2,−3)$

Answer:

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

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

315) $[T] xy+\sin(x)=1,(\frac{π}{2},0)$

Answer:

$y=0$

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

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

b. Find the equation of the normal line to the tangent line in a. at the point $(2,1)$.

317) For the equation $x^2+2xy−3y^2=0,$

a. Find the equation of the normal to the tangent line at the point $(1,1)$.

b. At what other point does the normal line in a. intersect the graph of the equation?

Answer:

$a. y=−x+2 b. (3,−1)$

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

319) For the equation $x^2+xy+y^2=7$,

a. Find the $x$-intercept(s).

b.Find the slope of the tangent line(s) at the x-intercept(s).

c. What does the value(s) in b. indicate about the tangent line(s)?

Answer:

$a. (±7√,0) b. −2$ c. They are parallel since the slope is the same at both intercepts.

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

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

Answer:

$y=−x+1$

322) Find $y′$ and $y''$ for $x^2+6xy−2y^2=3$.

323) [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 $60x^{3/4}y^{1/4}=3240$.

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

b. Interpret the result of a.

Answer:

$a. −0.5926$ b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.

324) [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 $30x^{1/3}y^{2/3}=360$.

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

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

b. Interpret the result of a.

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

Answer:

$−8$

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

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

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

Answer:

$−2.67$