I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation
$$ z(t+a) = h(a)z(t) $$
he makes the following derivation (I'm citing the book):
If we differentiate both sides of [the aforementioned equation] with respect to $a$, we obtain $$ z'(t+a) = h'(a)z(t). $$ Setting $a = 0$ gives $$ z'(t) = Hz(t) $$ where $$ H \equiv h'(0). $$ This implies $$ z(t) \propto e^{Ht}. $$ Thus the irreducible solution is an exponential! [...]
How can he differentiate with respect to $a$ then solve the resulting differential equation with respect to $t$? This makes no sense to me.