# Write formulas for the indicated partial derivatives for the multivariable function. g(k, m) = k^4m^5 − 3km a)g_k b)g_m c)g_m|_(k=2)

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

### Relevant Questions

Write formulas for the indicated partial derivatives for the multivariable function.
$$\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{3}}{m}^{{6}}−{8}{k}{m}$$
a)$$\displaystyle{g}_{{k}}$$
b)$$\displaystyle{g}_{{m}}$$
c)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$
Write formulas for the indicated partial derivatives for the multivariable function.
$$\displaystyle{k}{\left({a},{b}\right)}={3}{a}{b}^{{4}}+{8}{\left({1.4}^{{b}}\right)}$$
a) $$\displaystyle\frac{{\partial{k}}}{{\partial{a}}}$$
b) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}$$
c) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}$$
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)(delf)/(dely)ZSK
c)$$\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}$$
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)$$\displaystyle{g}_{{z}}$$
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: $$\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}$$.
Write first and second partial derivatives
$$\displaystyle{g{{\left({r},{t}\right)}}}={t}{\ln{{r}}}+{11}{r}{t}^{{7}}-{5}{\left({8}^{{r}}\right)}-{t}{r}$$
a)$$\displaystyle{g}_{{r}}$$
b)$$\displaystyle{g}_{{{r}{r}}}$$
c)$$\displaystyle{g}_{{{r}{t}}}$$
d)$$\displaystyle{g}_{{t}}$$
e)$$\displaystyle{g}_{{{\mathtt}}}$$
Write first and second partial derivatives
$$\displaystyle{f{{\left({x},{y}\right)}}}={2}{x}{y}+{9}{x}^{{2}}{y}^{{3}}+{7}{e}^{{{2}{y}}}+{16}$$
a)$$\displaystyle{f}_{{x}}$$
b)$$\displaystyle{f}_{{\times}}$$
c)$$\displaystyle{f}_{{{x}{y}}}$$
d)$$\displaystyle{f}_{{y}}$$
e)$$\displaystyle{f}_{{{y}{y}}}$$
f)$$\displaystyle{f}_{{{y}{x}}}$$
$$\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}$$
a) Calculate $$\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}$$
Let $$\displaystyle{z}{\left({x},{y}\right)}={e}^{{{3}{x}{y}}},{x}{\left({p},{q}\right)}=\frac{{p}}{{q}}{\quad\text{and}\quad}{y}{\left({p},{q}\right)}=\frac{{q}}{{p}}$$ are functions. Use multivariable chain rule of partial derivatives to find
(i) $$\displaystyle\frac{{\partial{z}}}{{\partial{p}}}$$
(ii) $$\displaystyle\frac{{\partial{z}}}{{\partial{q}}}$$.
Average value over a multivariable function using triple integrals. Find the average value of $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}+{y}^{{2}}+{z}^{{2}}$$ over the cube in the first octant bounded bt the coordinate planes and the planes x=5, y=5, and z=5