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If E Y = Y X , Prove that D Y D X = ( Log Y ) 2 Log Y − 1 ? - CBSE (Arts) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

Question

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

Solution

$\text{ We have }, e^y = y^x$

Taking log on both sides,

$\log e^y = \log y^x$
$\Rightarrow y \log e = x \log y$
$\Rightarrow y = x \log y . . . \left( i \right)$

Differentiating with respect to x,

$\frac{dy}{dx} = \frac{d}{dx}\left( x \log y \right)$
$\Rightarrow \frac{dy}{dx} = x\frac{dy}{dx}\left( \log y \right) + \log y\frac{d}{dx}\left( x \right)$
$\Rightarrow \frac{dy}{dx} = \frac{x}{y}\frac{dy}{dx} + \log y$
$\Rightarrow \frac{dy}{dx}\left( 1 - \frac{x}{y} \right) = \log y$
$\Rightarrow \frac{dy}{dx}\left( \frac{y - x}{y} \right) = \log y$
$\Rightarrow \frac{dy}{dx} = \frac{y \log \ y}{y - x}$
$\Rightarrow \frac{dy}{dx} = \frac{y \log \ y}{\left( y - \frac{y}{\log \ y} \right)} \left[ \text{ Using equation} \left( i \right) \right]$
$\Rightarrow \frac{dy}{dx} = \frac{y \log y\left( \log y \right)}{y \log \ y - y}$
$\Rightarrow \frac{dy}{dx} = \frac{y \left( \log y \right)^2}{y\left( \log \ y - 1 \right)}$
$\Rightarrow \frac{dy}{dx} = \frac{\left( \log y \right)^2}{\left( \log y - 1 \right)}$

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