CBSE (Arts) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

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

Login
Create free account


      Forgot password?

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,

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

  Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

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.
S
View in app×