Question

# Find all the second-order partial derivatives of the functions g(x,y)=\cos x^{2}-\sin 3y

Derivatives
Find all the second-order partial derivatives of the functions $$\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}$$

2021-05-19
Step 1: Given that
Find all the second-order partial derivatives of the functions $$\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}$$
Step 2: Finding the Second Order Partial Derivative
$$\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}$$
$$\displaystyle{\frac{{\partial}}{{\partial{x}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left({{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\right)}=-{\sin{{\left({x}^{{{2}}}\right)}}}\times{2}{x}-{0}=-{2}{x}{\sin{{\left({x}^{{{2}}}\right)}}}$$
$$\displaystyle{\frac{{\partial^{{{2}}}}}{{\partial{x}^{{{2}}}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{y}}}}{\left({{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\right)}=-{3}{\cos{{3}}}{y}$$
$$\displaystyle{\frac{{\partial^{{{2}}}}}{{\partial{y}^{{{2}}}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left(-{3}{\cos{{3}}}{y}\right)}={9}{\sin{{3}}}{y}$$