Partial Derivatives

(requires JavaScript)

1. Find the first partial derivatives of the function

$f\left(x,y\right)=\frac{x-y}{x+y}$

${f}_{x}\left(x,y\right)=2y/{\left(x+y\right)}^{2}$, ${f}_{y}\left(x,y\right)=-2x/{\left(x+y\right)}^{2}$
2. Find the first partial derivatives of the function

$f\left(r,s\right)=r\mathrm{ln}\left({r}^{2}+{s}^{2}\right)$

${f}_{r}\left(r,s\right)=\frac{2{r}^{2}}{{r}^{2}+{s}^{2}}+\mathrm{ln}\left({r}^{2}+{s}^{2}\right)$, ${f}_{s}\left(r,s\right)=\frac{2rs}{{r}^{2}+{s}^{2}}$
3. Find the first partial derivatives of the function

$f\left(x,y,z\right)={x}^{2}{e}^{yz}$

4. Find ${f}_{y}\left(-6,4\right)$ if $f\left(x,y\right)=\mathrm{sin}\left(2x+3y\right)$.
5. Verify that the conclusion of Clairaut's Theorem holds for the function $u\left(x,y\right)={x}^{4}{y}^{2}-2x{y}^{5}$.
6. Find all second order partial derivatives of the function

$z=f\left(x,y\right)=x{e}^{xy}$

${f}_{xx}\left(x,y\right)=2y{e}^{xy}+x{y}^{2}{e}^{xy}$,
${f}_{yy}\left(x,y\right)={x}^{3}{e}^{xy}$,
${f}_{xy}\left(x,y\right)={f}_{yx}\left(x,y\right)=2x{e}^{xy}+{x}^{2}y{e}^{xy}$.