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प्रश्न
Using differential, find the approximate value of the following: \[\left( 0 . 009 \right)^\frac{1}{3}\]
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उत्तर
\[\text { Consider the function } y = f\left( x \right) = \sqrt[3]{x} . \]
\[\text { Let }: \]
\[ x = 0 . 008\]
\[x + ∆ x = 0 . 009\]
\[\text { Then }, ∆ x = 0 . 001\]
\[\text { For } x = 0 . 008, \]
\[ y = \sqrt{0 . 008} = 0 . 2\]
\[\text { Let }: \]
\[ dx = ∆ x = 0 . 001\]
\[\text { Now,} y = \sqrt[3]{x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{3 \left( x \right)^\frac{2}{3}}\]
\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 0 . 008} = \frac{1}{3 \times 0 . 04} = \frac{1}{0 . 12}\]
\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{0 . 12} \times 0 . 001 = \frac{1}{120}\]
\[ \Rightarrow ∆ y = \frac{1}{120} = 0 . 008333\]
\[ \therefore \left( 0 . 009 \right)^\frac{1}{3} = y + ∆ y = 0 . 208333\]
