Answer
$\displaystyle \frac{dy}{dx}=-\frac{x}{y}$
Work Step by Step
Differentiating an equation containing
terms of form $f(x)$ and $g(y)$,
terms $f(x)$ are differentiated directly, $\displaystyle \frac{d}{dx}[f(x)],$
and
terms $g(y)$ are differentiated using the chain rule,
$\displaystyle \frac{d}{dx}[g(y)]=\frac{d}{dy}[g(y)]\cdot\frac{dy}{dx}$
------------------
LHS: sum, second term: chain rule
RHS: constant
$x^{2}+y^{2}=5$
$2x+2y\displaystyle \cdot\frac{dy}{dx}=0\qquad-2x$
$2y\displaystyle \cdot\frac{dy}{dx}=-2x\qquad/\div 2y$
$\displaystyle \frac{dy}{dx}=\frac{-2x}{2y}=-\frac{x}{y}$
