Question:
Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following:
$x^{5}+y^{5}=5 x y$
Solution:
We are given with an equation $x^{5}+y^{5}=5 x y$, 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,
$5 x^{4}+5 y^{4} \frac{d y}{d x}=5\left[y(1)+x \frac{d y}{d x}\right]$
$\frac{d y}{d x}\left[y^{4}-x\right]=y-x^{4}$
$\frac{d y}{d x}=\frac{y-x^{4}}{y^{4}-x}$