Answer
$\displaystyle \frac{dy}{dx}=-\frac{2}{3}x.$
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,
$\displaystyle \frac{d}{dx}[g(y)]=\frac{d}{dy}[g(y)]\cdot\frac{dy}{dx}$
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$\displaystyle \frac{d}{dx}(3y+x^{2})=\frac{d}{dx}(5)$
$3\displaystyle \frac{dy}{dx}+2x=0\qquad /-2x$
$3\displaystyle \frac{dy}{dx}=-2x\qquad/\div(3)$
$\displaystyle \frac{dy}{dx}=-\frac{2}{3}x.$
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Alternatively, solving the initial equation for y:
$3y+x^{2}=5\quad/-x^{2}$
$3y=-x^{2}+5\quad/\div 3$
$y=-\displaystyle \frac{1}{3}x^{2}-3$
Differentiating (constant mult., power, constant),
$\displaystyle \frac{dy}{dx}=-\frac{1}{3}\cdot 2x-0$
$\displaystyle \frac{dy}{dx}=-\frac{2}{3}x.$
