Derivative of a^x proof

Hey! I was recently trying to prove the derivative of a^x.

After trying it myself I ended up having to Google it because I don't seem to get it quite right. Here is what I tried to do:

$y = {a^{x}}$

$ln(y) = x*ln(a)$

$\frac{d}{dx} ln(y) = \frac{d}{dx} x*ln(a)$

Since ln(a) is a constant I just take the drivative of x which is one.

$\frac{1}{y} = ln(a)$

From above we see that y = a^x so I substitute it in there.

$\frac{1}{a^{x}}= ln(a)$

Rearranging gives me:

$1= ln(a) * a^{x}$

Here comes my problem, the result is supposed to be:

$\frac{dy}{dx}= ln(a)*a^{x}$

the dy/dx is supposed to come up on the left hand side as I take the derivative of ln(y) and get 1/y. I don't understand why this happens and I haven't seen any explanation behind for that specific step. Everything else is logical, just logarithmic/exponential properties. I would appreciate any help, thanks in advance!