Integrating $\frac{df}{f}$ I know the general solution to $\dfrac{df}{dt}=f$ is $f(t)=e^tC$. But how does one compute it from integrating both sides of $\dfrac{df}{f}=dt$? The left part does not seem to make any sense!

"Integrating" on  both sides gives you $$\int \frac 1 f df = \int 1 dt \\ \ln (f) = t + C \\ f = e^{t+C} = e^Ce^t$$ Note that this is an abuse of notation, you can't multiply by dt.
