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    i created a thread on the chain rule, bumped it today. created a new thread because i figured threads with 0 replies are ones which people looks at even if the OP hasnt had their question answered.

    i am stuck on these questions of showing things are true via the chain rule

    show by the chain rule that,

    1.dx/dt = 1/t(dx/ds)

    2. d^2x/dt^2 = 1/t^2(d^2x/ds^2) - 1/t^2(dx/ds)

    you are told that t = e^s (or s = lnt)


    for the first one, I got dx/dt = dx/ds * ds/dt, and ds/dt is differentiating lnt, so it's 1/t. so i did the first one correctly.

    however, the second one, i differentiated the first one using the product rule. I got d^2x/dt^2 = 1/t(d^2x/ds^2) - 1/t^2(dx/ds)

    I am missing out a t^2 on the bottom of the first fraction (the bold bit is wrong). i cant think of how to get a t^2 there. any ideas where i have gone wrong? thanks:o:
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    When you differentiated the second time, what did you differentiate with respect to? I think you've sort of done duv/dt = vdu/ds + udv/dt.
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    (Original post by IrrationalNumber)
    When you differentiated the second time, what did you differentiate with respect to? I think you've sort of done duv/dt = vdu/ds + udv/dt.
    i just followed the product rule, ignoring what i was differentiating with respect to. how do you do it, i have been stuck for ages on this thing:o:
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    2. d^2x/dt^2 = 1/t^2(d^2x/ds^2) - 1/t^2(dx/ds)
    You just forgot to get d/dt in terms of ds

     \frac{d^2}{dt^2} = \frac{d}{dt} [\frac{1}{t} \frac{dx}{ds}]

    Using product rule:

     \frac{d}{dt} [\frac{1}{t} \frac{dx}{ds}] = \frac{-1}{t^2} \frac {dx}{ds} + \frac{1}{t} \frac{d}{dt}[\frac{dx}{ds}]

    Now looking at the relationship for t and s we see that

     \frac{ds}{dt} = \frac{1}{t}

    so therefore

     dt = t ds

    This implies that

     \frac{d}{dt} = \frac{1}{t} \frac{d}{ds}

    Substitute this in the earlier expression to get

      \frac{d^2}{dt^2} = \frac{1}{t^2} \frac{d}{ds}[\frac{dx}{ds}] - \frac{1}{t^2} \frac{dx}{ds}

    Which gives the desired result.
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    (Original post by wrooru)
    i just followed the product rule, ignoring what i was differentiating with respect to. how do you do it, i have been stuck for ages on this thing:o:
    Well that's why it went wrong, you have to consider what you're differentiating with respect to.

    duv/dx = vdu/dx + udv/dx. You can't infer from that that
    duv/dt=vdu/ds + udv/dt... It should be duv/dt=vdu/dt+udv/dt. You can then use the chain rule on du/dt to get du/ds * ds/dt...
 
 
 
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