Using differentials, find the approximate values of the following… - Vidyadip

Question

Using differentials, find the approximate values of the following:
$\Big(\frac{17}{81}\Big)^{\frac{1}{4}}$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=(\text{x})\frac{1}{4}$
Let:
$\text{x}=\frac{16}{81}$
$\text{x}+\triangle \text{x}=\frac{17}{81}$
Then,
$\triangle\text{x}=\frac{1}{81}$
For $\text{x}=\frac{16}{81},$
$\text{y}=\Big(\frac{16}{81}\Big)^{\frac{1}{4}}=\frac{2}{3}$
Let:
$\text{dx}=\triangle \text{x}=\frac{1}{81}$
Now, $\text{y}=(\text{x})^ {\frac{1}{4}}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\frac{1}{4(\text{x})^{\frac{3}{4}}}$
$\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}=\frac{16}{81}}=\frac{27}{32}$
$\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=\frac{1} {12}\times\frac{1}{81}=\frac{1}{96}=0.01042$
$\Rightarrow\triangle \text{y} =0.01042$
$\therefore\Big(\frac{17}{81}\Big)^{\frac{1}{4}}=\text{y}+\triangle\text{y} =0.6771$

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