Question

# Find the four second partial derivatives of the following functions. Q(r,s)=\frac{e^{r^{e}s}}{s}

Derivatives
Find the four second partial derivatives of the following functions.
$$\displaystyle{Q}{\left({r},{s}\right)}={\frac{{{e}^{{{r}^{{{e}}}{s}}}}}{{{s}}}}$$

2021-03-25
Step 1
Partial differential :
$$\displaystyle{Q}{\left({r},{s}\right)}={\frac{{{e}^{{{r}^{{{e}}}{s}}}}}{{{s}}}}$$
$$\displaystyle{\frac{{\partial{Q}}}{{\partial{r}}}}={\frac{{{e}{r}^{{{2}}}{s}{e}^{{{r}^{{{3}}}{s}}}}}{{{s}}}}$$
$$\displaystyle={3}{e}^{{{r}^{{{3}}}{s}}}{r}^{{{2}}}$$
$$\displaystyle{\frac{{\partial{Q}}}{{\partial{r}}}}={\frac{{{s}.{r}^{{{3}}}{e}^{{{r}^{{{3}}}{s}}}-{e}^{{{r}^{{{3}}}{s}}}}}{{{s}^{{{2}}}}}}$$
Step 2
$$\displaystyle{\frac{{\partial^{{{2}}}{Q}}}{{\partial{r}^{{{2}}}}}}={3}{\left({3}{e}^{{{r}^{{{3}}}{s}}}{r}^{{{4}}}{s}+{2}{e}^{{{r}^{{{3}}}{s}}}{r}\right)}$$
$$\displaystyle{\frac{{\partial^{{{2}}}{Q}}}{{\partial{s}^{{{2}}}}}}={\frac{{{e}^{{{r}^{{{3}}}{s}}}{\left({r}^{{{6}}}-{s}^{{{2}}}-{2}{\left({r}^{{{3}}}{s}-{1}\right)}\right)}}}{{{s}^{{{3}}}}}}$$
$$\displaystyle{\frac{{\partial^{{{2}}}{Q}}}{{\partial{r}\partial{s}}}}={3}{e}^{{{r}^{{{3}}}{s}}}{r}^{{{5}}}={\frac{{\partial^{{{2}}}{Q}}}{{\partial{s}\partial{r}}}}$$
Here are the four second partial derivatives of the function.