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Thread: Need help finding the derivative.

  1. #1
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    I need to find the derivative of

    $\displaystyle x-2 \sqrt{x-1}$

    I knew to use the power rule, but I was coming up with
    $\displaystyle 1/2(x-1)^{-1/2}$ Is this the correct answer?

    Thanks
    Jason
    Last edited by mr fantastic; Jun 22nd 2009 at 01:03 AM. Reason: Merged posts, fixed latex
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Darkhrse99 View Post
    I knew to use the power rule, but I was coming up with
    $\displaystyle 1/2(x-1)^-1/2$ Is this the correct answer?
    The derivative (by power rule & chain rule) is $\displaystyle 1-\left(x-1\right)^{-1/2}=1-\frac{1}{\sqrt{x-1}}$, where the derivative of $\displaystyle \sqrt{x-1}$ is $\displaystyle \frac{1}{2}\left(x-1\right)^{-1/2}=\frac{1}{2\sqrt{x-1}}$
    Last edited by mr fantastic; Jun 22nd 2009 at 01:04 AM. Reason: Deleted some bits due to original ambiguity in OP
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    The derivative (by power rule & chain rule) is $\displaystyle 1-\left(x-1\right)^{-1/2}=1-\frac{1}{\sqrt{x-1}}$, where the derivative of $\displaystyle \sqrt{x-1}$ is $\displaystyle \frac{1}{2}\left(x-1\right)^{-1/2}=\frac{1}{2\sqrt{x-1}}$
    How did you know to use the chain rule?
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  4. #4
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    chain rule is present at any moment.

    for example if you want to differentiate a simple function as $\displaystyle x^2,$ then that is $\displaystyle 2x\cdot(x)'=2x,$ but that it's quite simple and then irrelevant to write, but having $\displaystyle \sqrt{x-1}$ we can think this as the composition by letting $\displaystyle f(x)=\sqrt x$ and $\displaystyle g(x)=x-1,$ so by differentiating $\displaystyle f\big(g(x)\big)$ we get $\displaystyle f'\big(g(x)\big)\cdot g'(x).$ According to this we have that $\displaystyle \sqrt{x-1}$ can be expressed as $\displaystyle h(x)=f\big(g(x)\big)=\sqrt{x-1},$ and then by differentiating we get (following up on what I said before) $\displaystyle h'(x)=\frac1{2\sqrt{x-1}}\cdot(x-1)'=\frac1{2\sqrt{x-1}}.$
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