Question:
Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following:
$(x+y)^{2}=2 a x y$
Solution:
We are given with an equation $(x+y)^{2}=2 a x y$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get,
$2(x+y)\left(1+\frac{d y}{d x}\right)=2 a\left[y+x \frac{d y}{d x}\right]$
$x+y+\frac{d y}{d x}[x+y]=a\left[y+x \frac{d y}{d x}\right]$
$\frac{d y}{d x}[x+y-a x]=a y-x-y$
$\frac{d y}{d x}=\frac{y(a-1)-x}{y+x(1-a)}$