Question

# Find the four second partial derivatives of the following functions. F(r,s)=re^{s}

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

2021-05-21
Given,
$$\displaystyle{F}{\left({r},{s}\right)}={r}{e}^{{{s}}}$$
Partially differentiating with respect to r, we get
$$\displaystyle{F}_{{{r}}}={e}^{{{s}}}\cdot{1}$$
$$\displaystyle={e}^{{{s}}}$$
Partially differentiating with respect to s, we get
$$\displaystyle{F}_{{{s}}}={r}{e}^{{{s}}}$$
Step 2
Now,
$$\displaystyle{F}_{{{r}{r}}}={\frac{{\partial{\left({F}_{{{r}}}\right)}}}{{\partial{r}}}}$$
$$\displaystyle={\frac{{\partial{\left({e}^{{{s}}}\right)}}}{{\partial{r}}}}$$
=0
$$\displaystyle{F}_{{{r}{s}}}={\frac{{\partial{\left({F}_{{{r}}}\right)}}}{{\partial{s}}}}$$
$$\displaystyle={\frac{{\partial{\left({e}^{{{s}}}\right)}}}{{\partial{s}}}}$$
$$\displaystyle={e}^{{{s}}}$$
$$\displaystyle{F}_{{{r}{s}}}={\frac{{\partial{\left({F}_{{{s}}}\right)}}}{{\partial{r}}}}$$
$$\displaystyle={\frac{{\partial{\left({r}{e}^{{{s}}}\right)}}}{{\partial{r}}}}$$
$$\displaystyle={e}^{{{s}}}$$
$$\displaystyle{F}_{{{s}{s}}}={\frac{{\partial{\left({F}_{{{s}}}\right)}}}{{\partial{s}}}}$$
$$\displaystyle={\frac{{\partial{\left({r}{e}^{{{s}}}\right)}}}{{\partial{s}}}}$$
$$\displaystyle={r}{e}^{{{s}}}$$