# Write first and second partial derivativesf(x,y)=2xy+9x^2y^3+7e^(2y)+16a)f_xb)f_(xx)c)f_(xy)d)f_ye)f_(yy)f)f_(yx)

Multivariable functions

Write first and second partial derivatives
$$\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}$$
a)$$\displaystyle{f}_{{x}}$$
b)$$\displaystyle{f}_{{xx}}$$
c)$$\displaystyle{f}_{{{x}{y}}}$$
d)$$\displaystyle{f}_{{y}}$$
e)$$\displaystyle{f}_{{{y}{y}}}$$
f)$$\displaystyle{f}_{{{y}{x}}}$$

2021-02-03

a) $$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}\right]}$$
$$\displaystyle{f}_{{x}}={2}{y}+{18}{x}{y}^{{3}}+{0}+{0}={2}{y}+{18}{x}{y}^{{3}}$$
$$\displaystyle\Rightarrow{f}_{{x}}={2}{y}+{18}{x}{y}^{{3}}$$
b) $$\displaystyle{f}_{{xx}}=\frac{{\partial{f}_{{x}}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{9}{y}+{18}{x}{y}^{{3}}\right]}$$
$$\displaystyle={0}+{18}{y}^{{3}}={18}{y}^{{3}}$$
$$\displaystyle{f}_{{xx}}={18}{y}^{{3}}$$
с) $$\displaystyle{f}_{{{x}{y}}}=\frac{{\partial{f}_{{x}}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{y}+{18}{x}{y}^{{3}}\right]}$$
$$\displaystyle{f}_{{{x}{y}}}={2}+{18}{x}\cdot{3}{y}^{{2}}$$
$$\displaystyle{f}_{{{x}{y}}}={2}+{54}{x}{6}^{{2}}$$
d) $$\displaystyle{f}_{{y}}=\frac{{\partial{f}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{4}{e}^{{{2}{y}}}+{16}\right]}$$
$$\displaystyle{f}_{{y}}={2}{x}+{9}{x}^{{2}}\cdot{3}{y}^{{2}}+{7}\cdot{2}{e}^{{{2}{y}}}+{0}$$
$$\displaystyle{f}_{{y}}={9}{x}+{27}{x}^{{2}}{y}+{14}{e}^{{2}}{y}$$
e) $$\displaystyle{f}_{{{y}{y}}}=\frac{\partial}{{\partial{y}}}{\left[{2}{x}+{27}{x}^{{2}}{y}^{{2}}+{14}{e}^{{{2}{y}}}\right.}$$
$$=0+27x^{2}x^{y^{2}}+14\cdot2e^{2y}$$
$$\displaystyle{f}_{{y}{y}}={54}{x}^{{2}}\cdot{2}{y}+{28}{e}^{{{2}{y}}}$$
f) $$f_{yx}=\frac{\partial f_{y}}{\partial{x}}=\frac{\partial}{\partial{x}}[2x+27x^{y^{2}}+14e^{2y}]$$
$$\displaystyle{f}_{{y}{y}}={2}={54}{x}{y}^{{2}}$$