Question

Compute all of the second-order partial derivatives for the functions and show that the mixed partial derivatives are equal. f(x,y)=e^{x}\sin(xy)

Derivatives
ANSWERED
asked 2021-05-12
Compute all of the second-order partial derivatives for the functions and show that the mixed partial derivatives are equal.
\(\displaystyle{f{{\left({x},{y}\right)}}}={e}^{{{x}}}{\sin{{\left({x}{y}\right)}}}\)

Answers (1)

2021-05-14
Step 1
Given function is:
\(\displaystyle{f{{\left({x},{y}\right)}}}={e}^{{{x}}}{\sin{{\left({x}{y}\right)}}}\)
Differentiating f(x, y) partially with respect to x we get,
\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={e}^{{{x}}}{\sin{{\left({x}{y}\right)}}}+{y}{e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}={e}^{{{x}}}{\left[{\sin{{\left({x}{y}\right)}}}+{y}{\cos{{\left({x}{y}\right)}}}\right]}\)
Differentiating f(x, y) partially with respect to y we get,
\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={x}{e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}\)
Step 2
Moreover we get,
\(\displaystyle{{f}_{{{x}{y}}}{\left({x},{y}\right)}}={x}{e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}+{e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}-{x}{y}{e}^{{{x}}}{\sin{{\left({x}{y}\right)}}}\)
\(\displaystyle{{f}_{{{y}{x}}}{\left({x},{y}\right)}}={e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}+{x}{e}^{{{x}}}{\cos{{\left({x}{y}\right)}}}-{x}{y}{e}^{{{x}}}{\sin{{\left({x}{y}\right)}}}\)
These are the second order partial derivatives of f(x, y).
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