Question

Write formulas for the indicated partial derivatives for the multivariable function.f(x,y)=7x^2+9xy+4y^3a)(delf)/(delx)b)(delf)/(dely)c)(delf)/(delx)|_(y=9)

Multivariable functions
ANSWERED
asked 2021-03-18

Write formulas for the indicated partial derivatives for the multivariable function.
\(\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\)
a)\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}\)
b) \(\frac{\partial f}{\partial y}\)
c)\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}\)

Expert Answers (1)

2021-03-19
\(\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\) (1)
a) Differentiating with respect to x partially, we have
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}=\frac{\partial}{{\partial{x}}}{\left[{7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\right]}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={7}\cdot{2}{x}+{9}{y}+{0}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}={14}{x}+{9}{y}\)
b) Differentiating with respect to y partially, we have
\(\displaystyle\frac{{\partial{f}}}{{\partial{y}}}=\frac{\partial}{{\partial{y}}}{\left[{7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\right]}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{y}}}={0}+{9}{x}+{4}\cdot{3}{y}^{{2}}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{y}}}={9}{x}+{12}{y}^{{2}}\)
c) \(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}={14}{x}+{9}\cdot{9}\)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}={14}{x}+{81}\)
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