Question

Find the partial derivatives, fx and fy, for f(x,y)=8x^{2}-6xy^{2}+3y+21

Derivatives
ANSWERED
asked 2021-05-04
Find the partial derivatives, fx and fy, for \(\displaystyle{f{{\left({x},{y}\right)}}}={8}{x}^{{{2}}}-{6}{x}{y}^{{{2}}}+{3}{y}+{21}\)

Answers (1)

2021-05-05
Step 1
Given:
\(\displaystyle{f{{\left({x},{y}\right)}}}={8}{x}^{{{2}}}-{6}{x}{y}^{{{2}}}+{3}{y}+{21}\)
Step 2
To find:
\(\displaystyle{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{x}}}},{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{y}}}}\)
Step 3
Calculation:
\(\displaystyle{f{{\left({x},{y}\right)}}}={8}{x}^{{{2}}}-{6}{x}{y}^{{{2}}}+{3}{y}+{21}\)
\(\displaystyle{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({8}{x}^{{{2}}}-{6}{x}{y}^{{{2}}}+{3}{y}+{21}\right)}\)
\(\displaystyle={16}{x}-{6}{y}^{{{2}}}\)
\(\displaystyle{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({8}{x}^{{{2}}}-{6}{x}{y}^{{{2}}}+{3}{y}+{21}\right)}\)
\(\displaystyle=-{12}{x}{y}+{3}\) Step 4 Answer: \(\displaystyle{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{x}}}}={16}{x}-{6}{y}^{{{2}}}\)
\(\displaystyle{\frac{{\partial{f{{\left({x},{y}\right)}}}}}{{\partial{y}}}}=-{12}{x}{y}+{3}\)
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