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

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.

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  • $\begingroup$ 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 $\endgroup$
    – Cato
    Feb 16, 2017 at 17:05
  • $\begingroup$ Not every differentiation rule has a name. This is just the chain rule in play. $\endgroup$ Feb 16, 2017 at 17:09
  • $\begingroup$ 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 $\endgroup$
    – Triatticus
    Feb 16, 2017 at 17:32

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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: $$[\ln(f(x))]'=\ln'(f(x))\cdot f'(x)=\frac 1{f(x)}\cdot f'(x)=\frac{f'(x)}{f(x)}.$$

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This technique is called "logarithmic differentiation". The identity that you have is just a result of the chain rule. Let $f = f(t)$, then

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

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