Answer
$\frac{dy}{dt}$ = $\frac{(-3t^{2})(t^{2}+4)}{(t^{2}-4)^{4}}$
Work Step by Step
$y$ = $(\frac{t^{2}}{t^{3}-4t})^{3}$
$\frac{dy}{dt}$ = $3(\frac{t^{2}}{t^{3}-4t})^{2}$ $\frac{d}{dt}$$(\frac{t^{2}}{t^{3}-4t})$
$\frac{dy}{dt}$ = $3(\frac{t^{2}}{t^{3}-4t})^{2}$ $\frac{(t^{3}-4t)\frac{d}{dt}(t^{2})-(t^{2})\frac{d}{dt}(t^{3}-4t)}{(t^{3}-4t)^{2}}$
$\frac{dy}{dt}$ = $3(\frac{t^{2}}{t^{3}-4t})^{2}$ $\frac{(t^{3}-4t)(2t)-(t^{2})(3t^{2}-4)}{(t^{3}-4t)^{2}}$
$\frac{dy}{dt}$ = $3(\frac{t^{2}}{t^{3}-4t})^{2}$ $\frac{(2t^{4}-8t^{2}-3t^{4}+4t^{2})}{(t^{3}-4t)^{2}}$
$\frac{dy}{dt}$ = $3(\frac{t^{2}}{t^{3}-4t})^{2}$ $\frac{(-t^{4}-4t^{2})}{(t^{3}-4t)^{2}}$
$\frac{dy}{dt}$ = $\frac{3t^{4}}{(t^{3}-4t)^{2}}$ $\frac{(-t^{2})(t^{2}+4)}{(t^{3}-4t)^{2}}$
$\frac{dy}{dt}$ = $\frac{(-3t^{6})(t^{2}+4)}{(t^{3}-4t)^{4}}$
$\frac{dy}{dt}$ = $\frac{(-3t^{6})(t^{2}+4)}{(t^{4})(t^{2}-4)^{4}}$
$\frac{dy}{dt}$ = $\frac{(-3t^{2})(t^{2}+4)}{(t^{2}-4)^{4}}$
