Question

# Find the first partial derivatives of the following functions. f(x,y)=4x^{3}y^{2}+3x^{2}y^{3}+10

Derivatives
Find the first partial derivatives of the following functions.
$$\displaystyle{f{{\left({x},{y}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}$$

2021-05-04
Step 1
Given, $$\displaystyle{f{{\left({x},{y}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}$$
The partial derivative of f(x,y) with respect to x is,
$$\displaystyle{f}'{\left({x},{y}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}$$
$$\displaystyle={\frac{{\partial}}{{\partial{x}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}\right)}+{\frac{{\partial}}{{\partial{x}}}}{\left({3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}$$
$$\displaystyle={12}{y}^{{{2}}}{x}^{{{2}}}+{y}^{{{3}}}{x}+{0}$$
$$\displaystyle={12}{y}^{{{2}}}{x}^{{{2}}}+{y}^{{{3}}}{x}$$
Step 2
Similarly,
The partial derivative of f(x,y) with respect to y is,
$$\displaystyle{f}'{\left({x},{y}\right)}={\frac{{\partial}}{{\partial{y}}}}{\left({4}{x}^{{{3}}}{y}^{{{2}}}+{3}{x}^{{{2}}}{y}^{{{3}}}+{10}\right)}$$
$$\displaystyle={\frac{{\partial}}{{\partial{y}}}}{\left({3}{x}^{{{2}}}{y}^{{{3}}}\right)}+{\frac{{\partial}}{{\partial{y}}}}{\left({10}\right)}$$
$$\displaystyle={8}{x}^{{{3}}}{y}+{9}{x}^{{{2}}}{y}^{{{2}}}+{0}$$
$$\displaystyle={8}{x}^{{{3}}}{y}+{9}{x}^{{{2}}}{y}^{{{2}}}$$