Find $\frac { d z } { d t }$ for $z = \frac { x } { y } , x = t ^ { 2 } , y = e ^ { t }$
A) $- 2 t e ^ { - t }$
B) $2 t e ^ { - t } - t ^ { 2 } e ^ { - t }$
C) $t e ^ { - t } - t ^ { 2 } e ^ { - 2 t }$
D) $2 t e ^ { - t }$

Question

Quizplus · Accepted Answer

Given $z = \frac{x}{y}$, with $x = t^2$ and $y = e^t$, we can differentiate $z$ with respect to $t$ using the quotient rule: $\frac{d}{dt}\left(\frac{x}{y}\right) = \frac{y\frac{dx}{dt} - x\frac{dy}{dt}}{y^2}$. Substituting $x = t^2$, $y = e^t$, $\frac{dx}{dt} = 2t$, and $\frac{dy}{dt} = e^t$, we get $\frac{d}{dt}\left(\frac{t^2}{e^t}\right) = \frac{e^t(2t) - t^2(e^t)}{e^{2t}} = 2te^{-t} - t^2e^{-t}$.

Find dzdt\frac { d z } { d t }dtdz​ For z=xy,x=t2,y=etz = \frac { x } { y } , x = t ^ { 2 } , y = e ^ { t }z=yx​,x=t2,y=et

Find $\frac { d z } { d t }$ For $z = \frac { x } { y } , x = t ^ { 2 } , y = e ^ { t }$