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Math Help - Help understanding the chain rule to solve a differential equation

  1. #1
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    Help understanding the chain rule to solve a differential equation

    I am following an example on a text book where the author is solving the next equation:

    dp/dt = (p-900)/2

    or if p is different than 900,

    1/(p-900)dp/dt = 1/2 (so far so good for me)

    But here the author states that by the chain rule the right hand side of the last equation is the derivative of ln|p-900| with respect to t:

    d/dt (ln|p-900|) = 1/2

    I really do not understand this last step. Any help will be very much appreciated
    The reference of the book is: Boyce, DiPrima "Elementary Differential Equations"
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  2. #2
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    Re: Help understanding the chain rule to solve a differential equation

    If you were to differentiate \displaystyle \begin{align*} \ln{\left| p - 900 \right|} \end{align*} with respect to t, remembering that p is a function of t, we have

    \displaystyle \begin{align*} \frac{d}{dt} \left( \ln{ \left| p - 900 \right| } \right) &= \frac{d}{dp} \left( \ln{ \left| p - 900 \right| } \right) \frac{dp}{dt} \\ &= \left( \frac{1}{p - 900} \right) \frac{dp}{dt}  \end{align*}
    Thanks from matemauch
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    Re: Help understanding the chain rule to solve a differential equation

    Now it is clear for me. Thanks "Prove it"
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