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

Derivatives
ANSWERED Find both first partial derivatives. $$\displaystyle{z}={e}^{{{x}{y}}}$$ 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, $$\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={y}{e}^{{{x}{y}}} \ {\frac{{\partial{z}}}{{\partial{y}}}}={x}{e}^{{{x}{y}}}$$