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प्रश्न
Using differential, find the approximate value of the \[{25}^\frac{1}{3}\] ?
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उत्तर
\[\text { Consider the function }y = f\left( x \right) = \left( x \right)^\frac{1}{3} . \]
\[\text {Let }: \]
\[ x = 27\]
\[ x + ∆ x = 25\]
\[\text { Then,} \]
\[ \bigtriangleup x = - 2\]
\[\text { For } x = 27, \]
\[ y = \left( 27 \right)^\frac{1}{3} = 3\]
\[\text { Let }: \]
\[ dx = ∆ x = - 2\]
\[\text { Now }, y = \left( x \right)^\frac{1}{3} \]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{3 \left( x \right)^\frac{2}{3}}\]
\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 27} = \frac{1}{27}\]
\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{27} \times \left( - 2 \right) = - 0 . 07407\]
\[ \Rightarrow ∆ y = - 0 . 07407\]
\[ \therefore \left( 25 \right)^\frac{1}{3} = y + ∆ y = 2 . 9259\]
