Can someone show me how to get the functional derivative of
$S[\varphi]=\int dx e^{-\alpha\varphi(x)}(\varphi'(x))^2$
In my notes I have the following:
But I don't get the steps especially the step from the second last to the last step. The one $\delta$ is misplaced, isn't it?
I think you're right; in the second last line, it should only say \begin{align} \int_{\Bbb{R}}dx \left\{\dots + e^{-\alpha \varphi(x)} \left(2 \varphi'(x) \dfrac{d}{dx}(\delta \varphi) \right) \right\} \end{align} In going from this step to the last, perform integration by parts on this last term. That's why you get a minus sign in front of $\dfrac{d}{dx}$ in the last line. You should also get a boundary term when you get integration by parts, but if you assume that $\delta \varphi$ is zero outside some interval $[-R,R]$, then the boundary term vanishes.