Question

Find both first partial derivatives. z = x^{2}e^{2y}

Derivatives
Find both first partial derivatives. $$\displaystyle{z}={x}^{{{2}}}{e}^{{{2}{y}}}$$

2021-04-13
Step 1
Given function is $$\displaystyle{z}={x}^{{{2}}}{e}^{{{2}{y}}}$$.
Partial derivative means derivative of function with respect to one variables keeping other variables as constant.
Finding partial derivative of the function with respect to x keeping y as constant.
$$\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}$$
$$\displaystyle={e}^{{{2}{y}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}\right)}$$
$$\displaystyle={e}^{{{2}{y}}}\cdot{2}{x}$$
$$\displaystyle={2}{{x}_{{{e}}}^{{{2}{y}}}}$$
Therefore, $$\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{{x}_{{{e}}}^{{{2}{y}}}}$$.
Step 2
Now, finding partial derivative with respect to y keeping other variable as constant.
$$\displaystyle{\frac{{\partial{z}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}$$
$$\displaystyle={x}^{{{2}}}\cdot{2}{e}^{{{2}{y}}}$$
$$\displaystyle={2}{x}^{{{2}}}{e}^{{{2}{y}}}$$
Hence, $$\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{x}{e}^{{{2}{y}}}{\quad\text{and}\quad}{\frac{{\partial{z}}}{{\partial{y}}}}={2}{x}{e}^{{{2}{y}}}$$.