Is the following true?
\begin{equation} \frac{\partial}{\partial (5a+3b)} \operatorname{Exp}[(5a+3b)\alpha - (3a+5b)\beta] = \alpha \operatorname{Exp}[(5a+3b)\alpha - (3a+5b)\beta] . \end{equation}
Can you differentiate with respect to a sum like this? (Or perhaps equivalently substitute $x=5a+3b$?). Note that $a$ and $b$ are not dependent on $\alpha$ or $\beta$, but all of the terms, $a$, $b$, $\alpha$ and $\beta$ are all dependent on a further variable, $\tau$. The equation above is a simplified version of what I actually want to find, which is a functional derivative.
Edit2: So my non-simplified question is whether this is true:
\begin{equation} \frac{\delta}{\delta x(\tau')} \operatorname{Exp}[\int_0^p d\tau (x (\tau) \alpha (\tau) - y (\tau)\beta (\tau))] = \alpha (\tau') \operatorname{Exp}[\int_0^p d\tau' (x (\tau')\alpha(\tau') - y (\tau')\beta (\tau'))] . \end{equation}
where $x (\tau)=5a (\tau)+3b (\tau)$, $y(\tau)=3a (\tau)+5b (\tau)$ and each of $\alpha$, $\beta$, $a$ and $b$ (and therefore $x$ and $y$) are operators. - Does it matter that the functional derivative is with respect to a sum?
Thanks.
Edit1: Thanks @henry for spotting my typo. - The right hand side of the equation is $\alpha \operatorname{Exp}[(5a+3b)\alpha - (3a+5b)\beta]$ not just $\alpha$!