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Derivative from First Principle

lim x → 027+x...

Question

$lim x \to 0 \frac{(27 + x)^{\frac{1}{3}} - 3}{9 - (27 + x)^{\frac{2}{3}}} =$

A

- \frac{1}{6}

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B

\frac{1}{6}

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C

\frac{1}{3}

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D

- \frac{1}{3}

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Solution

The correct option is A $- \frac{1}{6}$
$lim x \to 0 \frac{(27 + x)^{\frac{1}{3}} - 3}{9 - (27 + x)^{\frac{2}{3}}} (\frac{0}{0} form)$
Applying L Hosptial's rule
$= lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{\frac{1}{3} (27 + x)^{\frac{- 2}{3}}}{- \frac{2}{3} (27 + x)^{\frac{- 1}{3}}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - \frac{1}{6}$

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Derivative from First Principle

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