Question

Find the first and second derivatives of the following and simplify whenever possible:

x=e^t: y=te^-t

Answer 1

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{e^{-t}-te^{-t}}{e^t}=(1-t)e^{-2t}.$

$\frac{d^2y}{dx^2}=\frac{\frac{d^2y}{dt^2}\frac{dx}{dt}-\frac{dy}{dt}\frac{d^2x}{dt^2}}{(\frac{dx}{dt})^3}= \frac{(t-2)e^{-t}*e^t-(1-t)e^{-t}*e^t}{e^{3t}}=(2t-3)e^{-3t}.$