Question

Find $\frac{dy}{dx}$ in each of the following cases:

${\left(x+y\right)}^2=2axy$

Answer 1

$\mathrm{We}\mathrm{have},{\left(x+y\right)}^2=2axy$
Differentiating with respect to x, we get,
$$\begin{gathered} \Rightarrow \frac{d}{dx}{\left(x+y\right)}^2=\frac{d}{dx}\left(2axy\right) \\ \Rightarrow 2\left(x+y\right)\frac{d}{dx}\left(x+y\right)=2a\left[x\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+y\frac{d}{dx}\left(x\right)\right] \\ \Rightarrow 2\left(x+y\right)\left[1+\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right]=2a\left[x\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+y\left(1\right)\right] \\ \Rightarrow 2\left(x+y\right)+2\left(x+y\right)\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=2ax\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+2ay \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\left[2\left(x+y\right)-2ax\right]=2ay-2\left(x+y\right) \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{2\left[ay-x-y\right]}{2\left[x+y-ax\right]} \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\left(\frac{ay-x-y}{x+y-ax}\right) \end{gathered}$$