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

Question

Using differentials, find the approximate values of the following:
$(33)^{\frac{1}{5}}$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=(\text{x})^{\frac{1}{5}}$
Let:
$\text{x} =32$
$\text{x}+\triangle \text{x}=33$
Then,
$\triangle\text{x}=1$
For $\text{x}=33$
$\text{y}=(32)^{\frac{1}{5}}=2$
Let:
$\text{dx}=\triangle \text{x}=1$
Now, $\text{y}=(\text{x})^ {\frac{1}{5}}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\frac{1}{5(\text{x})^{\frac{4}{5}}}$
$\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x} =32}=\frac{1}{80}$
$\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=\frac{1} {80}\times(1)=0.0125$
$\Rightarrow\triangle \text{y} =0.0125$
$\therefore(33)^{\frac{1}{5}} =\text{y}+\triangle\text{y} =2.0125$

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