Using differentials, find the approximate values of the following…

Question

Using differentials, find the approximate values of the following:
$(0.007)^{\frac{1}{3}}$

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Answer

Let $\text{x}=0.008,\\\text{x}+\triangle\text{x}=0.007$
$\triangle\text{x}=0.007-0.008$
$\triangle\text{x}=-0.001$
Let $\text{y}=\text{x}^{\frac{1}{3}}$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{3(\text{x})^{\frac{2}{3}}}$
$\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}=\frac{1}{3(0.008)^{\frac{2}{3}}}$
$=\frac{100}{2}$
$\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}=8.333$
$\triangle\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\text{x}\Rightarrow0.008}\times\triangle\text{x}$
$=(8.333)(-0.001)$
$\triangle\text{y}=-0.008333$
$(0.007)^{\frac{1}{3}}=\text{y}+\triangle\text{y}$
$=\text{x}^{\frac{1}{3}}-0.008333$
$=(0.008)^{\frac{1}{3}}-0.008333$
$=0.2-0.008333$
$(0.007)^{\frac{1}{3}}=0.191667$

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