Question

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
ANSWERED
asked 2021-02-02

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)\(\displaystyle{f}_{{{y}{x}}}\)

Answers (1)

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) \(f_{yx}=\frac{\partial f_{y}}{\partial{x}}=\frac{\partial}{\partial{x}}[2x+27x^{y^{2}}+14e^{2y}]\)
\(\displaystyle{f}_{{y}{y}}={2}={54}{x}{y}^{{2}}\)

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