Answer
$\frac{dy}{dt}$ = $\frac{(sin^{\frac{1}{3}}t)(4tcost-3sint)}{3t^{2}}$
Work Step by Step
$y$ = $(t^{-\frac{3}{4}}sint)^{\frac{4}{3}}$
$\frac{dy}{dt}$ = $\frac{4}{3}$$(t^{-\frac{3}{4}}sint)^{\frac{1}{3}}$ $\frac{d}{dt}$$(t^{-\frac{3}{4}}sint)$
$\frac{dy}{dt}$ = $\frac{4}{3}$$(t^{-\frac{3}{4}}sint)^{\frac{1}{3}}$ [$(t^{-\frac{3}{4}})$$\frac{d}{dt}$$(sint)$ + $(sint)$$\frac{d}{dt}$$(t^{-\frac{3}{4}})$]
$\frac{dy}{dt}$ = $\frac{4}{3}$$(t^{-\frac{3}{4}}sint)^{\frac{1}{3}}$ [$(t^{-\frac{3}{4}})$$(cost)$ + $(sint)$$(-\frac{3}{4})$$(t^{-\frac{7}{4}})$]
$\frac{dy}{dt}$ = $\frac{4}{3}$$(t^{-\frac{1}{4}}sin^{\frac{1}{3}}t)$ [$\frac{4tcost-3sint}{4t^{\frac{7}{4}}}$]
$\frac{dy}{dt}$ = $\frac{(4sin^{\frac{1}{3}}t)}{3t^{\frac{1}{4}}}$ [$\frac{4tcost-3sint}{4t^{\frac{7}{4}}}$]
$\frac{dy}{dt}$ = ($sin^{\frac{1}{3}}t$) [$\frac{4tcost-3sint}{3t^{2}}$]
$\frac{dy}{dt}$ = $\frac{(sin^{\frac{1}{3}}t)(4tcost-3sint)}{3t^{2}}$
