Problem: how to evaluate $\int_0^y[-\frac{d}{dx}(\ln(f(x))]dx$?
If we would have had $\int[-\frac{d}{dx}(\ln(f(x))]dx$ then the integration and differentiation would have killed each other but since we have a definite integral that's not possible.
Using Leibniz' Rule is possible but makes it more difficult.
In the derivation in my textbook they seem to ignore the derivative in its entirety and immediately apply the Fundamental Theorem of Calculus, i.e. they write $\int_0^y[-\frac{d}{dx}(\ln(f(x))]dx=-(\ln(f(y)-\ln(f(0))$.
I seem to be overlooking something. What is the correct approach and interpretation here?