I was wondering if there is a general way to solve the functional equation
$$(\exists k)(f^\prime(x) = f(x+k))$$
I know that this is true for certain functions:
$$(e^{cx})^\prime = e^{c(x+\frac{\ln c}{c})}$$ $$\sin(x)^\prime=\sin(x+\tfrac{\pi}{2})$$
but I was wondering if these functions take a general form or are related to the eigenvalues of the second derivative, because both of these are.