Question

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

Derivatives
Find the first partial derivatives of the following functions.
$$\displaystyle{f{{\left({x},{y}\right)}}}={e}{x}^{{{2}}}{y}$$

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:
$$\displaystyle{\frac{{\partial{f}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({e}^{{{x}^{{{2}}}{y}}}\right)}$$
$$\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}{y}\right)}$$
$$\displaystyle={e}^{{{x}^{{{2}}}{y}}}\cdot{\left({x}^{{{2}}}\right)}$$
$$\displaystyle={x}^{{{2}}}{e}^{{{x}^{{{2}}}{y}}}$$