Question

Find  \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?

 

Answer 1

\[\text{ We have, } \left( x + y \right)^2 = 2axy\]

Differentiating with respect to x, we get,

\[\Rightarrow \frac{d}{dx} \left( x + y \right)^2 = \frac{d}{dx}\left( 2axy \right)\]
\[ \Rightarrow 2\left( x + y \right)\frac{d}{dx}\left( x + y \right) = 2a\left[ x\frac{d y}{d x} + y\frac{d}{dx}\left( x \right) \right] \]
\[ \Rightarrow 2\left( x + y \right)\left[ 1 + \frac{d y}{d x} \right] = 2a\left[ x\frac{d y}{d x} + y\left( 1 \right) \right]\]
\[ \Rightarrow 2\left( x + y \right) + 2\left( x + y \right)\frac{d y}{d x} = 2ax\frac{d y}{d x} + 2ay\]
\[ \Rightarrow \frac{d y}{d x}\left[ 2\left( x + y \right) - 2ax \right] = 2ay - 2\left( x + y \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{2\left[ ay - x - y \right]}{2\left[ x + y - ax \right]}\]
\[ \Rightarrow \frac{d y}{d x} = \left( \frac{ay - x - y}{x + y - ax} \right)\]