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
asked 2020-11-23
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) \(\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}\)

Answers (1)

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) \(\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}={12}\cdot{3}{b}^{{3}}+{8}{\ln{{\left({1.4}\right)}}}{\left({1.4}^{{b}}\right)}\)
\(\displaystyle\frac{{\partial{k}}}{{\partial{b}}}{\mid}_{{{a}={3}}}={36}{b}^{{3}}+{8}{\ln{{\left({1.4}\right)}}}{\left({1.4}^{{b}}\right)}\)
0

Relevant Questions

asked 2021-01-27
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)\(\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}\)
asked 2021-02-27
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)\(\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}\)
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)(delf)/(dely)ZSK
c)\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}\)
asked 2021-02-22
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)\(\displaystyle{g}_{{z}}\)
asked 2021-03-01
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)\(\displaystyle{g}_{{{\mathtt}}}\)
asked 2020-12-17
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.
asked 2021-02-02
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)\(\displaystyle{f}_{{{y}{x}}}\)
asked 2021-03-02
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: \(\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}\).
asked 2021-01-06
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) \(\displaystyle\frac{{\partial{z}}}{{\partial{q}}}\).
asked 2021-02-18
A surface is represented by the following multivariable function,
\(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{3}}+{y}^{{3}}-{3}{x}-{3}{y}+{1}\)
Calculate coordinates of stationary points.
...