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# If Y = Log ( √ X + 1 √ X ) Prove that D Y D X = X − 1 2 X ( X + 1 ) ? - CBSE (Commerce) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If $y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$prove that $\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}$ ?

#### Solution

$\text{ We have, y } = \log\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$

Differentiate it with respect to x

$\frac{d y}{d x} = \frac{d}{dx}\log\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$

$= \frac{1}{\sqrt{x} + \frac{1}{\sqrt{x}}}\frac{d}{dx}\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$

$= \frac{\sqrt{x}}{x + 1}\left( \frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}} \right)$

$= \frac{1}{2}\frac{\sqrt{x}}{x + 1}\left( \frac{x - 1}{x\sqrt{x}} \right)$

$= \frac{x - 1}{2x\left( x + 1 \right)}$

$So, \frac{d y}{d x} = \frac{x - 1}{2x\left( x + 1 \right)}$

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Solution If Y = Log ( √ X + 1 √ X ) Prove that D Y D X = X − 1 2 X ( X + 1 ) ? Concept: Simple Problems on Applications of Derivatives.
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