Question

Find the four second partial derivatives of the following functions. f(x,y)=x^{2}\sin y

Derivatives
ANSWERED
asked 2021-03-30
Find the four second partial derivatives of the following functions.
\(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{2}}}{\sin{{y}}}\)

Answers (1)

2021-04-01
Step 1
The given function is \(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{2}}}{\sin{{y}}}\).
This implies that,
\(\displaystyle{f}_{{{x}}}={2}{x}{\sin{{y}}}\)
\(\displaystyle{f}_{{{y}}}={x}^{{{2}}}{\cos{{y}}}\)
Step 2
Now obtain the second derivatives as follows.
\(\displaystyle{f}_{{\times}}={2}{\sin{{y}}}\)
\(\displaystyle{f}_{{{y}{y}}}=-{x}^{{{2}}}{\sin{{y}}}\)
\(\displaystyle{f}_{{{x}{y}}}={2}{x}{\cos{{y}}}\)
\(\displaystyle{f}_{{{y}{x}}}={2}{x}{\cos{{y}}}\)
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