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

Question

Using differentials, find the approximate values of the following:
$(0.009)^{\frac{1}{3}}$

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Answer

consider the function $=\text{f}(\text{x})=\sqrt[3]{\text{x}}$
Let:
x = 0.008
$\text{x}+\triangle\text{x}=0.009$
then, $\triangle\text{x}=0.001$
For x = 0.008,
$\text{y}=\sqrt{0.008}=0.2$
Let:
$\text{d}+\triangle\text{d}=0.001$
Now $=\text{y}=\sqrt[3]{\text{x}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{3(\text{x})^{\frac{2}{3}}}$
$\Rightarrow\Big(\frac{\text{dy}}{\text{d}}\Big)_{\text{x}=0.008}=\frac{1}{3\times0.04}-\frac{1}{0.12}$
$\therefore\triangle\text{y}=\frac{1}{120}=0.008333$
$\therefore(0.009)^{\frac{1}{3}}=\text{y}+\triangle\text{y}=0.208333$

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