3.8 Implicit differentiation

$x^2-y^2=4$

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

$x^{2}y=y-7$

$3x^3+9x{y}^2=5x^3$

$xy-\text{cos}\left(xy\right)=1$

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

$\text{−}xy-2=\frac{x}{7}$

$y\text{sin}\left(xy\right)=y^2+2$

${\left(xy\right)}^2+3x=y^2$

$x^{3}y+x{y}^3=\mathrm{-8}$

[T] $x^{4}y-x{y}^3=\mathrm{-2},\left(\mathrm{-1},\mathrm{-1}\right)$

[T] $x^2{y}^2+5xy=14,\left(2,1\right)$

[T] $\text{tan}\left(xy\right)=y,\left(\frac{\pi }{4},1\right)$

[T] $x{y}^2+\text{sin}\left(\pi y\right)-2x^2=10,\left(2,\mathrm{-3}\right)$

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

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

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

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

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

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

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

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

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

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

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

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

[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 $60x^{3\text{/}4}{y}^{1\text{/}4}=3240.$

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

[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 $30x^{1\text{/}3}{y}^{2\text{/}3}=360.$

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

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

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 $y=16.$

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

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

$x=\text{sin}y$

$x=\text{cos}y$

$x=\text{tan}y$