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?

score 2 · Answer 1 · answered May 13 '18 at 05:33

2

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 $\ln(|t|)+C$.

answered May 13 '18 at 05:33

Alex R.

32,771

I know the derivative but how the above equation(2) is derived? Can you explain it a bit more? – shashack May 13 '18 at 06:09
Take the derivative of $\ln|p-900|$. – Alex R. May 13 '18 at 19:13
I know the derivative as I said. I needed this kind of explanation https://math.stackexchange.com/questions/1252405/is-it-mathematically-valid-to-separate-variables-in-a-differential-equation – shashack May 14 '18 at 02:30
Thanks for paying attention my question anyway. – shashack May 14 '18 at 02:30

Chain rule example

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