Answer
$$\frac{dw}{dt}=-e^{-t}$$
Work Step by Step
(a) We will use Chain Rule:
$$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}=\frac{\partial}{\partial x}(xy)\frac{d}{dt}(e^t)+\frac{\partial}{\partial y}(xy)\frac{d}{dt}(e^{-2t})=y\cdot e^t+x\cdot e^{-2t}\frac{d}{dt}(-2t)=ye^t-2xe^{-2t}$$
Expressing this in terms of $t$ we have:
$$\frac{dw}{dt}=e^{-2t}\cdot e^t-2e^t\cdot e^{-2t}=-e^{-t}$$
(b) We will first convert $w$ to a function of $t$ and then we will differentiate.
$$w=xy=e^t\cdot e^{-2t}=e^{-t}$$
$$\frac{dw}{dt}=\frac{d}{dt}(e^{-t})=e^{-t}\frac{d}{dt}(-t)=e^{-t}\cdot(-1)=-e^{-t}$$
