Question

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

Derivatives
ANSWERED
asked 2021-05-19
Find the four second partial derivatives of the following functions.
\(\displaystyle{F}{\left({r},{s}\right)}={r}{e}^{{{s}}}\)

Answers (1)

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
\(\displaystyle{F}_{{{r}{s}}}={\frac{{\partial{\left({F}_{{{r}}}\right)}}}{{\partial{s}}}}\)
\(\displaystyle={\frac{{\partial{\left({e}^{{{s}}}\right)}}}{{\partial{s}}}}\)
\(\displaystyle={e}^{{{s}}}\)
\(\displaystyle{F}_{{{r}{s}}}={\frac{{\partial{\left({F}_{{{s}}}\right)}}}{{\partial{r}}}}\)
\(\displaystyle={\frac{{\partial{\left({r}{e}^{{{s}}}\right)}}}{{\partial{r}}}}\)
\(\displaystyle={e}^{{{s}}}\)
\(\displaystyle{F}_{{{s}{s}}}={\frac{{\partial{\left({F}_{{{s}}}\right)}}}{{\partial{s}}}}\)
\(\displaystyle={\frac{{\partial{\left({r}{e}^{{{s}}}\right)}}}{{\partial{s}}}}\)
\(\displaystyle={r}{e}^{{{s}}}\)
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