# Write formulas for the indicated partial derivatives for the multivariable function.g(k, m) = k^4m^5 − 3kma)g_kb)g_mc)g_m|_(k=2)

Multivariable functions

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)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$

2021-01-28

$$\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{4}}{m}^{{5}}−{3}{k}{m}$$ (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) $$\displaystyle{g}_{{m}}{\left|_{\left({k}={2}\right)}=\frac{{\partial{q}}}{{\partial{m}}}\right|}_{{{k}={2}}}={5}\cdot{\left({2}\right)}^{{4}}{m}^{{4}}-{3}\cdot{2}$$
$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}={80}{m}^{{4}}-{6}$$