Question

Using differentiation, find the approximate value of each of the following:

a) ${\frac{17}{81}}^{\frac{1}{4}}$

b) ${\left(33\right)}^{-\frac{1}{5}}$

Answer 1

$a)$ let $x=\frac{16}{81},△x=\frac{1}{81},f\left(x\right)=x^{\frac{1}{4}}f(x+△x)=f\left(x\right)+f^{\prime }\left(x\right)△xf\left(\frac{17}{81}\right)=\sqrt[4]{4\frac{16}{81}}+\frac{1}{4}(\frac{16}{81}{)}^{\frac{-3}{4}}\times \frac{1}{81}=0.677$

$b$) let $x=32,△x=1,f\left(x\right)=x^{\frac{-1}{5}}{f}^{\prime }\left(x\right)=\frac{-1}{5}f(x+△x)=f\left(x\right)+f^{\prime }\left(x\right)△xf\left(33\right)={\left(32\right)}^{\frac{-1}{5}}-\frac{{\left(32\right)}^{\frac{-6}{5}}}{5}=0.497$