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