# Write formulas for the indicated partial derivatives for the multivariable function. k(a,b)=3ab^4+8(1.4^b) a) (delk)/(dela) b) (delk)/(delb) c) (delk)/(delb)|_(a=3)

Question
Multivariable functions
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) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}$$

2020-11-24
a) $$\displaystyle\frac{{\partial{k}}}{{\partial{a}}}=\frac{\partial}{{\partial{a}}}{\left[{3}{a}{b}^{{4}}+{8}{\left({1.4}^{{b}}\right)}\right]}$$
$$\displaystyle\frac{{\partial{k}}}{{\partial{a}}}={3}{b}^{{4}}+{0}={3}{b}^{{4}}$$
$$\displaystyle\frac{{\partial{k}}}{{\partial{a}}}={3}{b}^{{4}}$$
b) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}=\frac{\partial}{{\partial{b}}}{\left[{3}{a}{b}^{{4}}+{8}{\left({1.4}^{{b}}\right)}\right]}$$
$$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}={3}{a}\cdot{4}{b}^{{9}}+{8}{\left({1.4}^{{b}}\right)}{\log{{\left({1}\cdot{4}\right)}}}$$
$$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}={12}{a}{b}^{{3}}+{8}{\ln{{\left({1.4}\right)}}}{\left({1.4}^{{b}}\right)}$$
c) $$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}={12}\cdot{3}{b}^{{3}}+{8}{\ln{{\left({1.4}\right)}}}{\left({1.4}^{{b}}\right)}$$
$$\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}={36}{b}^{{3}}+{8}{\ln{{\left({1.4}\right)}}}{\left({1.4}^{{b}}\right)}$$

### Relevant Questions

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)$$\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}$$
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.
$$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}$$
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)$$\displaystyle{g}_{{z}}$$
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)$$\displaystyle{g}_{{{\mathtt}}}$$
A surface is represented by the following multivariable function,
$$\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}$$
a) Calculate $$\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}$$
b) Calculate coordinates of stationary points.
c) Classify all stationary points.
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)$$\displaystyle{f}_{{{y}{x}}}$$
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: $$\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}$$.
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) $$\displaystyle\frac{{\partial{z}}}{{\partial{q}}}$$.
$$\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{3}}+{y}^{{3}}-{3}{x}-{3}{y}+{1}$$