Question

Write formulas for the indicated partial derivatives for the multivariable function.
\(\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{4}}{m}^{{5}}−{3}{k}{m}\)
a)\(\displaystyle{g}_{{k}}\)
b)\(\displaystyle{g}_{{m}}\)
c)\(\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}\)

Answer 1

\(\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{4}}{m}^{{5}}−{3}{k}{m}\) (1)
a) Differentiating with respect to x partially, we have
\(\displaystyle\frac{{\partial{g}}}{{\partial{k}}}=\frac{\partial}{{\partial{k}}}{\left[{k}^{{4}}{m}^{{5}}−{3}{k}{m}\right]}\)
\(\displaystyle\Rightarrow\frac{{\partial{g}}}{{\partial{k}}}={4}{k}^{{3}}{m}^{{5}}-{3}{m}\)
b) Differentiating with respect to x partially, we have
\(\displaystyle\frac{{\partial{g}}}{{\partial{m}}}=\frac{\partial}{{\partial{m}}}{\left[{k}^{{4}}{m}^{{5}}−{3}{k}{m}\right]}\)
\(\displaystyle\Rightarrow\frac{{\partial{g}}}{{\partial{m}}}={5}{k}^{{4}}{m}^{{4}}-{3}{k}\)
c) \(\displaystyle{g}_{{m}}{\left|_{\left({k}={2}\right)}=\frac{{\partial{q}}}{{\partial{m}}}\right|}_{{{k}={2}}}={5}\cdot{\left({2}\right)}^{{4}}{m}^{{4}}-{3}\cdot{2}\)
\(\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}={80}{m}^{{4}}-{6}\)