Chain rule example

Question

I have a question about chain rule example.

The below equation is from Elementary Differential Equations and Boundary Value Problems by William E.Boyce

$\frac{dp/dt}{p-900} = \frac{1}{2} \dots (1)$

$\frac{d}{dt}\ln\lvert p-900 \rvert = \frac{1}{2} \dots (2)$ by the chain rule

I have no idea of how to derive (1) -> (2) by the chain rule.

Can you please show how to derive?

Answer 1

By definition $\frac{d\ln(t)}{dt}=1/t$. For $\ln(|t|)$, it breaks into two cases. If $t>0$, then it's just $\ln(t)$ again. If $t<0$, then it's $\ln(-t)$, whose derivative is $-1/t$. In other words, the anti-derivative of $1/t$ is $\ln(|t|)+C$.