awful derivation of the time evolution operator

It’s a well-known fact that the Schrödinger-equation \(\mathrm{i}\hbar\partial_t\psi = H\psi\) defines a (unitary) time-evolution operator $U(t)$. Please spot all mistakes in the following derivation, which nevertheless yields the correct result! Let’s start by dividing both sides of teh equation by $\psi$, leading to \(\frac{\partial_t\psi}{\psi} = -\frac{\mathrm{i}H}{\hbar}\) followed by using the derivative property of the logarithm, \(\partial_t\ln\psi = -\frac{\mathrm{i}H}{\hbar}\) which can be integrated to be \(\ln\psi = -\frac{\mathrm{i}Ht}{\hbar}\) and exponentiated to \(\psi(t) \propto \exp\left(-\frac{\mathrm{i}Ht}{\hbar}\right).\) Oh my.