Question

# 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)

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