편도함수

Find the indicated higher-order partial derivatives.

Given $z={e}^{x}\tan \,y,$ find $\frac{{\partial }^{2}z}{\partial x\partial y}$ and $\frac{{\partial }^{2}z}{\partial y\partial x}.$

Find the indicated higher-order partial derivatives.

Given $f(x,y,z)=xyz,$ find ${f}_{xyy},{f}_{yxy},$ and ${f}_{yyx}.$

Find the indicated higher-order partial derivatives.

Find ${f}_{xx}(x,y)$ for $f(x,y)=\frac{4{x}^{2}}{y}+\frac{{y}^{2}}{2x}.$

Find the indicated higher-order partial derivatives.

Let $f(x,y,z)={x}^{2}{y}^{3}z-3x{y}^{2}{z}^{3}+5{x}^{2}z-{y}^{3}z.$ Find ${f}_{xyz}.$

Find the indicated higher-order partial derivatives.

Let $F(x,y,z)={x}^{3}y{z}^{2}-2{x}^{2}yz+3xz-2{y}^{3}z.$ Find ${F}_{xyz}.$

Find the indicated higher-order partial derivatives.

Given $f(x,y)={x}^{2}+x-3xy+{y}^{3}-5,$ find all points at which ${f}_{x}={f}_{y}=0$ simultaneously.

Given $f(x,y)=2{x}^{2}+2xy+{y}^{2}+2x-3,$ find all points at which $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$ simultaneously.

Given $f(x,y)={y}^{3}-3y{x}^{2}-3{y}^{2}-3{x}^{2}+1,$ find all points on $f$ at which ${f}_{x}={f}_{y}=0$ simultaneously.

Find the indicated higher-order partial derivatives.

Given $f(x,y)=15{x}^{3}-3xy+15{y}^{3},$ find all points at which ${f}_{x}(x,y)={f}_{y}(x,y)=0$ simultaneously.

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)=-7x-2xy+7y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta y\to 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}$ for $f(x,y)=-7x-2xy+7y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)={x}^{2}{y}^{2}+xy+y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)=\sin (xy).$

The law of cosines can be thought of as a function of three variables. Let $x,y,$ and $\theta$ be two sides of any triangle where the angle $\theta$ is the included angle between the two sides. Then, $F(x,y,\theta )={x}^{2}+{y}^{2}-2xy\,\cos \,\theta$ gives the square of the third side of the triangle. Find $\frac{\partial F}{\partial \theta }$ and $\frac{\partial F}{\partial x}$ when $x=2,y=3,$ and $\theta =\frac{\pi }{6}.$

A Cobb-Douglas production function is $f(x,y)=200{x}^{0.7}{y}^{0.3},$ where $x\,\text{and}\,y$ represent the amount of labor and capital available. Let $x=500$ and $y=1000.$ Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at these values, which represent the marginal productivity of labor and capital, respectively.

The domain of $f(x,y)={x}^{3}{\sin }^{-1}(y)$ is $x=$ all real numbers, and $-\pi \le y\le \pi .$

The linear approximation to the function of $f(x,y)=5{x}^{2}+x\,\tan (y)$ at $(2,\pi )$ is given by $L(x,y)=22+21(x-2)+(y-\pi ).$

For the following exercises, find the largest region of continuity for the function.

$f(x,y)={x}^{3}{\sin }^{-1}(y)$

For the following exercises, find the largest region of continuity for the function.

$g(x,y)=\ln (4-{x}^{2}-{y}^{2})$

For the following exercises, find all first derivatives, full or partial, as appropriate.

$f(x,y)=\sqrt{{x}^{2}-{y}^{2}}$