Question

Given f(x,y)=2x^{2}-xy^{3}+4y^{6}, findf_{xx}(x,y)=f_{xy}(x,y)=

Derivatives
ANSWERED
asked 2021-06-07

Given \(\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}^{{{2}}}-{x}{y}^{{{3}}}+{4}{y}^{{{6}}}\), find
\(\displaystyle{{f}_{{xx}}{\left({x},{y}\right)}}=\)
\(\displaystyle{{f}_{{{x}{y}}}{\left({x},{y}\right)}}=\)

Answers (1)

2021-06-08

Step 1
Consider the given function.
\(\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}^{{{2}}}-{x}{y}^{{{3}}}+{4}{y}^{{{6}}}\)
The objective of the question is to find the value of \(\displaystyle{{f}_{{xx}}{\left({x},{y}\right)}}\ {\quad\text{and}\quad}\ {{f}_{{{x}{y}}}{\left({x},{y}\right)}}\).
These are the second order partial derivatives.
Step 2
Partially differentiate the function with respect to x.
\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={2}\cdot{3}{x}^{{{3}-{1}}}-{1}\cdot{y}^{{{3}}}+{0}\)
\(\displaystyle={6}{x}^{{{2}}}-{y}^{{{3}}}\)
Partially differentiate again with respect to x to find \(\displaystyle{{f}_{{xx}}{\left({x},{y}\right)}}\).
\(\displaystyle{{f}_{{xx}}{\left({x},{y}\right)}}={6}\cdot{2}{x}-{0}={12}{x}\)
Partially differentiate the function with respect to y.
\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={0}-{x}\cdot{3}{y}^{{{3}-{1}}}+{4}\cdot{6}{y}^{{{6}-{1}}}\)
\(\displaystyle=-{3}{x}{y}^{{{2}}}+{24}{y}^{{{5}}}\)
Partially differentiate again with respect to x to find \(\displaystyle{{f}_{{{x}{y}}}{\left({x},{y}\right)}}\).
\(\displaystyle{{f}_{{{x}{y}}}{\left({x},{y}\right)}}=-{3}{y}^{{{2}}}\cdot{1}+{0}\)
\(\displaystyle=-{3}{y}^{{{2}}}\)
Thus, the required partial derivatives are obtained.
\(\displaystyle{{f}_{{xx}}{\left({x},{y}\right)}}={12}{x}\)
\(\displaystyle{{f}_{{{x}{y}}}{\left({x},{y}\right)}}=-{3}{y}^{{{2}}}\)

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