Answer
$\displaystyle \frac{dy}{dx}=\frac{4}{5}$
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,
$\displaystyle \frac{d}{dx}[g(y)]=\frac{d}{dy}[g(y)]\cdot\frac{dy}{dx}$
------------------
$\displaystyle \frac{d}{dx}(4x-5y)=\frac{d}{dx}(9)$
$4-5\displaystyle \frac{dy}{dx}=0\qquad /-4$
$-5\displaystyle \frac{dy}{dx}=-4\qquad/\div(-5)$
$\displaystyle \frac{dy}{dx}=\frac{4}{5}.$
========================
Alternatively, solving the initial equation for y:
$4x-5y=9\quad/-4x$
$-5y=-4x+9\quad/\div(-5)$
$y=\displaystyle \frac{4}{5}x+\frac{9}{5}$
Differentiating (constant multiple, constant),
$\displaystyle \frac{dy}{dx}=\frac{4}{5}+0=\frac{4}{5}$
