Question

COnsider the multivariable function g(x,y)=x^2-3y^4x^2+sin(xy). Find the following partial derivatives: g_x. g_y, g_(xy), g(xx), g(yy).

Multivariable functions
ANSWERED
asked 2021-03-02
COnsider the multivariable function \(\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{\left({x}{y}\right)}}}\). Find the following partial derivatives: \(\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}\).

Answers (1)

2021-03-03
\(\displaystyle{g}_{{x}}=\frac{\partial}{{\partial{x}}}{\left({x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{x}}}\right)}{)}\) [treat y as cons \(\displaystyle{\tan{{t}}}\)]
\(\displaystyle={2}{x}-{6}{x}{y}^{{4}}+{y}{\cos{{x}}}{y}\)
Again
\(\displaystyle{g}_{{y}}=\frac{\partial}{{\partial{y}}}{\left({x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{x}}}{y}\right)}\) [treat x as cons \(\displaystyle{\tan{{t}}}\)]
\(\displaystyle=-{12}{x}^{{2}}{y}^{{3}}+{x}{\cos{{x}}}{y}\)
Now
\(\displaystyle{g}_{{\times}}=\frac{\partial}{{\partial{x}}}{\left({g}_{{x}}\right)}=\frac{\partial}{{\partial{x}}}{\left({2}{x}-{6}{x}{y}^{{4}}+{y}{\cos{{x}}}{y}\right)}\)
\(\displaystyle={2}-{6}{y}^{{4}}-{y}^{{2}}{\sin{{x}}}{y}\)
Again
\(\displaystyle{g}_{{{y}{y}}}=\frac{\partial}{{\partial{y}}}{\left(-{12}{x}^{{2}}{y}^{{3}}+{x}{\cos{{x}}}{y}\right)}\)
\(\displaystyle=-{36}{x}^{{2}}{y}^{{2}}-{x}^{{2}}{\sin{{x}}}{y}\)
Also
\(\displaystyle{g}_{{{x}{y}}}=\frac{\partial}{{\partial{y}}}{\left({g}_{{x}}\right)}\)
\(\displaystyle=\frac{\partial}{{\partial{y}}}{\left({2}{x}-{6}{x}{y}^{{4}}+{y}{\cos{{x}}}{y}\right)}\)
\(\displaystyle=-{24}{x}{y}^{{3}}-{y}^{{2}}{\sin{{x}}}{y}+{\cos{{x}}}{y}\)
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