TITLE: derivative of a function divided by the same function
QUESTION [5 upvotes]: I've been trying to understand and look for a proof that for example
(1) $$\frac{\frac{d}{dx}f(x)}{f(x)}$$
is equal to
(2) $$\frac{d}{dx}ln[f(x)]$$
Can someone help me understand why 1 & 2 are equal?

REPLY [7 votes]: $$\frac{d}{dx}[\ln f(x)]= \frac{1}{f(x)} \cdot f'(x) = \frac{1}{f(x)}\cdot \frac{d}{dx}f(x) = \frac{\frac{d}{dx}f(x)}{f(x)}$$
where the second step is true by the chain rule.