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

Question

Using differentials, find the approximate values of the following:
$(82)^{\frac{1}{4}}$

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Answer

Let $\text{x}=81,\\\text{x}+ \triangle\text{x}=82$$\triangle\text{x}=82- 81$
$=1$
Let $\text{y}=\text{x}^ {\frac{1}{4}}$
$\frac{\text{dy}} {\text{dx}}=\frac{1}{4\text{x}^{\frac{3} {4}}}$
$\Big(\frac{\text{dy}} {\text{dx}}\Big)_{\text{x}=81}=\frac{1} {4(81)^{\frac{3}{4}}}$
$=\frac{1}{108}$
$=0.009259$
$\triangle\text{y}= \Big(\frac{\text{dy}}{\text{dx}}\Big)_ {\text{x}=81}\times(\triangle\text{x})$
$=(0.0008259)(1)$
$=0.009259$
$(82)^{\frac{1}{4}}= \text{y}+ \triangle\text{y}$
$=\text {x}^\frac{1} {4}+0.009259$
$=(81)\frac{1} {4}+0.009259$
$=(81)\frac{1} {4}+3.009259$

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