Question

Find the first partial derivatives of the following functions. f(x,y)=4x^{3}y^{2}+3x^{2}y^{3}+10

Derivatives
ANSWERED
asked 2021-05-03
Find the first partial derivatives of the following functions.
\(\displaystyle{f{{\left({x},{y}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\)

Expert Answers (1)

2021-05-04
Step 1
Given, \(\displaystyle{f{{\left({x},{y}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\)
The partial derivative of f(x,y) with respect to x is,
\(\displaystyle{f}'{\left({x},{y}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}\)
\(\displaystyle={\frac{{\partial}}{{\partial{x}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}\right)}+{\frac{{\partial}}{{\partial{x}}}}{\left({3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}\)
\(\displaystyle={12}{y}^{{{2}}}{x}^{{{2}}}+{y}^{{{3}}}{x}+{0}\)
\(\displaystyle={12}{y}^{{{2}}}{x}^{{{2}}}+{y}^{{{3}}}{x}\)
Step 2
Similarly,
The partial derivative of f(x,y) with respect to y is,
\(\displaystyle{f}'{\left({x},{y}\right)}={\frac{{\partial}}{{\partial{y}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}\)
\(\displaystyle={\frac{{\partial}}{{\partial{y}}}}{\left({3}{x}^{{{2}}}{y}^{{{3}}}\right)}+{\frac{{\partial}}{{\partial{y}}}}{\left({10}\right)}\)
\(\displaystyle={8}{x}^{{{3}}}{y}+{9}{x}^{{{2}}}{y}^{{{2}}}+{0}\)
\(\displaystyle={8}{x}^{{{3}}}{y}+{9}{x}^{{{2}}}{y}^{{{2}}}\)
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