Question

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

Derivatives
ANSWERED
asked 2021-02-22
Find the first partial derivatives of the following functions.
\(\displaystyle{f{{\left({x},{y}\right)}}}={e}{x}^{{{2}}}{y}\)

Answers (1)

2021-02-24
Step 1
Obtain the first partial derivative of f with respect to x as shown below:
\(\displaystyle{\frac{{\partial{f}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({e}^{{{x}^{{{2}}}{y}}}\right)}\)
\(\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}{y}\right)}\)
\(\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\left({2}{x}{y}\right)}\)
\(\displaystyle={2}{x}{y}{e}^{{{x}^{{{2}}}{y}}}\)
Step 2
Obtain the first partial derivative of f with respect to y as shown below:
\(\displaystyle{\frac{{\partial{f}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({e}^{{{x}^{{{2}}}{y}}}\right)}\)
\(\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}{y}\right)}\)
\(\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\left({x}^{{{2}}}\right)}\)
\(\displaystyle={x}^{{{2}}}{e}^{{{x}^{{{2}}}{y}}}\)
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