Answer
$$\frac{\partial W}{\partial E} =-\frac{1}{kT}e^{-E/kT},$$
$$\frac{\partial W}{\partial T} =\frac{E}{kT^2}e^{-E/kT}.$$
Work Step by Step
Since $ W=e^{-E/kT}$, then using the chain rule, we have
$$\frac{\partial W}{\partial E}=e^{-E/kT} (-1/kT)=-\frac{1}{kT}e^{-E/kT},$$
$$\frac{\partial W}{\partial T}=e^{-E/kT} (E/kT^2)=\frac{E}{kT^2}e^{-E/kT}.$$
