Question

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

Derivatives
ANSWERED
asked 2021-03-23
Find the four second partial derivatives of the following functions.
\(\displaystyle{Q}{\left({r},{s}\right)}={\frac{{{e}^{{{r}^{{{e}}}{s}}}}}{{{s}}}}\)

Answers (1)

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.
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