Question

Find the four second partial derivatives of the following functions. f(x,y)=x^{2}\sin y

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

2021-04-01
Step 1
The given function is $$\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{2}}}{\sin{{y}}}$$.
This implies that,
$$\displaystyle{f}_{{{x}}}={2}{x}{\sin{{y}}}$$
$$\displaystyle{f}_{{{y}}}={x}^{{{2}}}{\cos{{y}}}$$
Step 2
Now obtain the second derivatives as follows.
$$\displaystyle{f}_{{\times}}={2}{\sin{{y}}}$$
$$\displaystyle{f}_{{{y}{y}}}=-{x}^{{{2}}}{\sin{{y}}}$$
$$\displaystyle{f}_{{{x}{y}}}={2}{x}{\cos{{y}}}$$
$$\displaystyle{f}_{{{y}{x}}}={2}{x}{\cos{{y}}}$$