Question

Write formulas for the indicated partial derivatives for the multivariable function. g(x,y,z)=3.1x^2yz^2+2.7x^y+z a)g_x b)g_y c)g_z

Multivariable functions
ANSWERED
asked 2021-02-22
Write formulas for the indicated partial derivatives for the multivariable function. \(\displaystyle{g{{\left({x},{y},{z}\right)}}}={3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}\)
a)\(\displaystyle{g}_{{x}}\)
b)\(\displaystyle{g}_{{y}}\)
c)\(\displaystyle{g}_{{z}}\)

Answers (1)

2021-02-23
a) \(\displaystyle\frac{{\partial{g}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}\right]}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{x}}}={\left({3.1}\right)}\cdot{2}{x}{y}{z}^{{2}}+{2.7}{y}{x}^{{{y}-{1}}}+{0}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{x}}}={6}\cdot{2}{x}{y}{z}^{{2}}+{2.7}{y}{x}^{{{y}-{1}}}\)
b) \(\displaystyle\frac{{\partial{g}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}\right]}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{y}}}={3.1}{x}^{{2}}{z}^{{2}}+{2.7}{x}^{{y}}{\ln{{\left({x}\right)}}}+{0}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{y}}}={3.1}{x}^{{2}}{z}^{{2}}+{2.7}{x}^{{y}}{\ln{{\left({x}\right)}}}\)
c)\(\displaystyle\frac{{\partial{g}}}{{\partial{z}}}=\frac{\partial}{{\partial{z}}}{\left[{3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}\right]}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{z}}}={3.1}{x}^{{2}}{y}{x}{\left({2}{z}\right)}+{0}+{1}\)
\(\displaystyle\frac{{\partial{g}}}{{\partial{z}}}={6}\cdot{2}{x}^{{2}}{y}{z}+{1}\)
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