Question

# Find the four second partial derivatives of the following functions. h(x,y)=x^{3}+xy^{2}+1

Derivatives
Find the four second partial derivatives of the following functions.
$$\displaystyle{h}{\left({x},{y}\right)}={x}^{{{3}}}+{x}{y}^{{{2}}}+{1}$$

2021-06-05
Step 1
Given information:
$$\displaystyle{h}{\left({x},{y}\right)}={x}^{{{3}}}+{x}{y}^{{{2}}}+{1}$$
Step 2
First find the partial derivative of the given function,
$$\displaystyle{\frac{{\partial{h}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{3}}}+{x}{y}^{{{2}}}+{1}\right)}$$
$$\displaystyle={3}{x}^{{{2}}}+{y}^{{{2}}}+{0}$$
$$\displaystyle={3}{x}^{{{2}}}+{y}^{{{2}}}$$
Now find the required double partial derivative of given function,
$$\displaystyle{\frac{{\partial^{{{2}}}{h}}}{{\partial{x}^{{{2}}}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({3}{x}^{{{2}}}+{y}^{{{2}}}\right)}$$
=6x+0+0
=6x