# Identify this Derivation Rule

$$\frac{d(\ln(f(t))}{dt} = \frac{f'(t)}{f(t)}$$

A Finnish introductory book on differential equations uses the above formula without explanation, simply claiming "it is known". What is this equation and why does it hold true for all f? I went through the entire "derivation rules" wikipedia page and could not find it. Searching for the specific formula is hard due to markdown.

• To derive the expression, use the chain rule along with the fact that d/dx(ln(x)) = 1/x : as a technique for integration, I've heard it called the quotient rule I believe
– Cato
Feb 16, 2017 at 17:05
• Not every differentiation rule has a name. This is just the chain rule in play. Feb 16, 2017 at 17:09
• While it's really just the chain rule at play, you can also call it the logarithmic derivative of f. It's very useful I'm differentiating complex rational expressions Feb 16, 2017 at 17:32

Assuming you know the following both rules (namely derivative of logarithm and chain rule) $$\ln'(x)=\frac 1x,\qquad [f(g(x))]'=f'(g(x))\cdot g'(x),$$ your rule is just a combination of both: $$[\ln(f(x))]'=\ln'(f(x))\cdot f'(x)=\frac 1{f(x)}\cdot f'(x)=\frac{f'(x)}{f(x)}.$$
This technique is called "logarithmic differentiation". The identity that you have is just a result of the chain rule. Let $f = f(t)$, then
$$\frac{d}{dt}(\ln f(t)) = \frac{d}{dt}(\ln f) \\ = \frac{d}{df}(\ln f)\cdot \frac{df}{dt} \\ = \frac{f'}{f} \\ = \frac{f'(t)}{f(t)}.$$