Find all the second-order partial derivatives of the functions f(x,y)=\sin xy

Derivatives
Find all the second-order partial derivatives of the functions $$\displaystyle{f{{\left({x},{y}\right)}}}={\sin{{x}}}{y}$$

2021-06-07

Step 1
the given function:
$$\displaystyle{f{{\left({x},{y}\right)}}}={\sin{{x}}}{y}$$
we have to find the second order partial derivatives of the given function:
i.e. $$\displaystyle{f}_{{xx}},{f}_{{{y}{y}}},{f}_{{{x}{y}}},{f}_{{{y}{x}}}$$.
Step 2
here, $$\displaystyle{f{{\left({x},{y}\right)}}}={\sin{{\left({x}{y}\right)}}}$$
firstly , we find $$\displaystyle{f}_{{{x}}},{f}_{{{y}}}$$
$$\displaystyle{f}_{{{x}}}={\frac{{\partial{f}}}{{\partial{x}}}}={y}{\cos{{x}}}{y}\ {\quad\text{and}\quad}\ {f}_{{{y}}}={\frac{{\partial{f}}}{{\partial{y}}}}={x}{\cos{{x}}}{y}$$
now we have to find $$\displaystyle{f}_{{xx}},{f}_{{{y}{y}}}$$
$$\displaystyle{f}_{{xx}}={\frac{{\partial{f}_{{{x}}}}}{{\partial{x}}}}=-{y}^{{{2}}}{\sin{{x}}}{y}\ {\quad\text{and}\quad}\ {f}_{{{y}{y}}}={\frac{{\partial{f}_{{{y}}}}}{{\partial{f}_{{{y}}}}}}=-{x}^{{{2}}}{\sin{{x}}}{y}$$
now we have to find $$\displaystyle{f}_{{{x}{y}}},{f}_{{{y}{x}}}$$
$$\displaystyle{f}_{{{x}{y}}}={\frac{{\partial{f}_{{{x}}}}}{{\partial{y}}}}=-{y}{x}{\sin{{x}}}{y}\ {\quad\text{and}\quad}\ {f}_{{{y}{x}}}={\frac{{\partial{f}_{{{y}}}}}{{\partial{x}}}}=-{x}{y}{\sin{{x}}}{y}$$
this is the required answer.