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Thread: Square root derivative

  1. #1
    Super Member Bacterius's Avatar
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    Square root derivative

    Hi, I'm trying to derivate this function : $\displaystyle f(x) = \sqrt{2x - 1}$. I'm applying this principle : if $\displaystyle f = a^n$ then $\displaystyle f' = na^{n - 1}$. So there I go :

    $\displaystyle f(x) = \sqrt{2x - 1}$

    $\displaystyle f(x) = (2x - 1)^{\frac{1}{2}}$

    Now I derivate :

    $\displaystyle f'(x) = \frac{1}{2} (2x - 1)^{\frac{-1}{2}}$

    $\displaystyle f'(x) = \frac{1}{2} \frac{1}{(2x - 1)^{\frac{1}{2}}}$

    $\displaystyle f'(x) = \frac{1}{2} \frac{1}{\sqrt{2x - 1}}$

    $\displaystyle f'(x) = \frac{1}{2 \sqrt{2x - 1}}$

    But the book says that's not right and the answer should be :

    $\displaystyle f'(x) = \frac{2}{2 \sqrt{2x - 1}} = \frac{1}{\sqrt{2x - 1}}$

    Does anyone see when I went wrong ? Thanks a lot ...
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by Bacterius View Post
    Hi, I'm trying to derivate this function : $\displaystyle f(x) = \sqrt{2x - 1}$. I'm applying this principle : if $\displaystyle f = a^n$ then $\displaystyle f' = na^{n - 1}$. So there I go :

    $\displaystyle f(x) = \sqrt{2x - 1}$

    $\displaystyle f(x) = (2x - 1)^{\frac{1}{2}}$

    Now I derivate :

    $\displaystyle f'(x) = \frac{1}{2} (2x - 1)^{\frac{-1}{2}}$ <<<<<<< here! You forgot to apply the chain rule

    $\displaystyle f'(x) = \frac{1}{2} \frac{1}{(2x - 1)^{\frac{1}{2}}}$

    $\displaystyle f'(x) = \frac{1}{2} \frac{1}{\sqrt{2x - 1}}$

    $\displaystyle f'(x) = \frac{1}{2 \sqrt{2x - 1}}$

    But the book says that's not right and the answer should be :

    $\displaystyle f'(x) = \frac{2}{2 \sqrt{2x - 1}} = \frac{1}{\sqrt{2x - 1}}$

    Does anyone see when I went wrong ? Thanks a lot ...
    $\displaystyle f(x) = (2x - 1)^{\frac{1}{2}}$

    $\displaystyle f'(x) = \underbrace{\frac{1}{2} (2x - 1)^{\frac{-1}{2}}}_{\text{der. of the square-root}} \cdot \underbrace{2}_{\text{der. of the radicand}}$
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  3. #3
    Super Member Bacterius's Avatar
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    Thanks a lot !
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