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Math Help - Finding derivative with the given information

  1. #1
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    Finding derivative with the given information

    \displaystyle y=f(\sqrt{x^2+9}) and \displaystyle f'(5)=-2, find \displaystyle\frac{dy}{dx} when \displaystyle x=4

    How would I start this question? Greatly appreciate the help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Use the Chain's Rule to y=f(\sqrt{x^2+9}) and substitute x=4 .


    Fernando Revilla
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    Quote Originally Posted by FernandoRevilla View Post
    Use the Chain's Rule to y=f(\sqrt{x^2+9}) and substitute x=4 .


    Fernando Revilla
    Substitute \displaystyle x=4 in the derivative of \displaystyle\sqrt{x^2+9}?
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    Answer in the back of the book is \displaystyle -\frac{8}{5}. I'm not sure where to begin.
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    MHF Contributor FernandoRevilla's Avatar
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    \dfrac{dy}{dx}=f'(\sqrt{x^2+9})\cdot \dfrac{x}{\sqrt{x^2+9}}

    Substitute x=4 .


    Fernando Revilla
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    Quote Originally Posted by FernandoRevilla View Post
    \dfrac{dy}{dx}=f'(\sqrt{x^2+9})\cdot \dfrac{x}{\sqrt{x^2+9}}

    Substitute x=4 .


    Fernando Revilla
    I'm not sure if I understand how f'(\sqrt{x^2+9}) works.

    Is  f(x)=\sqr{x^2+9}? If so, where do we use the  f'(5)=-2 information?
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  7. #7
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by youngb11 View Post
    I'm not sure if I understand how f'(\sqrt{x^2+9}) works.

    \dfrac{dy}{dx}(4)=f'(\sqrt {4^2+9})\cdot \dfrac{4}{\sqrt {4^2+9}}=f'(5)\cdot \dfrac{4}{5}=(-2)\cdot \dfrac {4}{5}=-\dfrac{8}{5}


    Fernando Revilla
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