Time derivative of operator I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger propagator, but I have some problems with the fact that $f$ depends on $t$ too.

Since the laplacian is a linear differential operator it can be differentiated like any other linear operator. The way the differential works is the same way as for the product of functions : Let $D(t)$ be a linear differential operator then $$D(t+\epsilon)f(x,t+\epsilon)-D(t)f(x,t)=$$$$D(t+\epsilon)f(x,t+\epsilon)-D(t)f(x,t+\epsilon)+D(t)f(x,t+\epsilon)-D(t)f(x,t)$$ Dividing by $ \epsilon$ and taking the limit $\lim_{t \rightarrow 0}$ then yields: $$\frac{\mathrm{d}}{\mathrm{d}t}(D(t)f(x,t))=\left(\frac{\mathrm{d}}{\mathrm{d}t}D(t)\right)f(x,t)+D(t)\left(\frac{\mathrm{d}}{\mathrm{d}t}f(x,t)\right)$$