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I Chain rule confusion

  1. Mar 28, 2016 #1
    while solving differential equations, I got a bit confused with chain rule problem.
    The solution says below
    yprime = z
    then
    y double prime = z (dz/dy) = z prime
    but I don't understand why the differentiation of z is in that form.
    Please help...
     
  2. jcsd
  3. Mar 28, 2016 #2

    pasmith

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    Homework Helper

    If [itex]y' = z[/itex] then, by the concept of equality and the definition of the second derivative, [itex]y'' = z'[/itex]. The chain rule has nothing to do with this.

    In kinematics, where [itex]v = \frac{ds}{dt}[/itex] and [itex]a = \frac{dv}{dt} = \frac{d^2 s}{dt^2}[/itex], then the chain rule gives [tex]
    a = \frac{dv}{dt} = \frac{dv}{ds} \frac{ds}{dt} = v \frac{dv}{ds},[/tex] a change of variable which is occasionally useful, particularly if [itex]a[/itex] is given in terms of [itex]s[/itex].
     
  4. Mar 28, 2016 #3

    Ssnow

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    Gold Member

    If ##y'=z## denote the independent variable with ##x## then ##y'(x)=z(x)## and ## y''(x)=\frac{d}{dx} y'(x)=\frac{d}{dx}z(x)=z'(x)##
     
  5. Mar 28, 2016 #4
    sorry but, that's not what i'm asking..
    I mean the second term in the equation.
     
    Last edited: Mar 28, 2016
  6. Mar 28, 2016 #5

    Mark44

    Staff: Mentor

    Are you asking about the part in the middle in the last equation?
    This doesn't make sense to me. The tacit assumption here seems to be that you're differentiating with respect to z, with z being the independent variable. What you have in the middle should be ##\frac d {dy}z##, which is different from ##z(\frac{dz}{dy})##.

    It would help if you showed us the actual problem.
     
  7. Mar 28, 2016 #6

    Ssnow

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    yes there is a problem with the notations as @Mark44 said, are you sure that the middle term is ##z\left(\frac{dz}{dy}\right)## ?
     
  8. Mar 28, 2016 #7
    i definitely agree with what you guys said and that was the reason why I was asking this here. Then do you think it's just a typo? I thought i mistook something.
     
  9. Mar 28, 2016 #8
    I found out why!! It is in fact, chain rule.
    if we differentiate left and right side by t
    dz/dt = dz/dy X dy/dt
    , dy/dt=y'=z...
     
  10. Mar 28, 2016 #9

    Ssnow

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    ok it is ##y''=\frac{d}{dx}y'=\frac{d}{dx}z=\frac{dy}{dx}\frac{dz}{dy}=z\frac{dz}{dy}##
     
  11. Mar 28, 2016 #10
    nice timing! lol we almost uploaded the response at the same time! anyway thanks a lot for sparing your time for my question! :)
     
  12. Mar 28, 2016 #11

    Ssnow

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    nothing! yes simultaneously. I was also in doubt at the beginning, this the miracle of calculations ...
     
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