Question

Find both first partial derivatives. z=e^{xy}

Derivatives
ANSWERED
asked 2021-04-09
Find both first partial derivatives. \(\displaystyle{z}={e}^{{{x}{y}}}\)

Answers (1)

2021-04-11

Step 1
Given function is \(\displaystyle{z}={e}^{{{x}{y}}}\).
Partial derivative of function means derivative of function with respect to one variable keeping other variable as constant.
Partial derivative of given function with respect to x keeping y as constant.
\(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({e}^{{{x}{y}}}\right)}\)
\(\displaystyle={e}^{{{x}{y}}}{\frac{{\partial}}{{\partial{x}}}}{\left({x}{y}\right)}\)
\(\displaystyle={e}^{{{x}{y}}}{\left({1}\cdot{y}\right)}\)
\(\displaystyle={y}{e}^{{{x}{y}}}\)
Step 2
Partial derivative of given function with respect to y keeping x as constant.
\(\displaystyle{\frac{{\partial{z}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({e}^{{{x}{y}}}\right)}\)
\(\displaystyle={e}^{{{x}{y}}}{\frac{{\partial}}{{\partial{y}}}}{\left({x}{y}\right)}\)
\(\displaystyle={e}^{{{x}{y}}}{\left({x}\cdot{1}\right)}\)
\(\displaystyle={x}{e}^{{{x}{y}}}\)
Hence, \(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={y}{e}^{{{x}{y}}} \ {\frac{{\partial{z}}}{{\partial{y}}}}={x}{e}^{{{x}{y}}}\)

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