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Thread: Finding derivatives

  1. #1
    Senior Member I-Think's Avatar
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    Finding derivatives

    Check my work please

    Find $\displaystyle f'(0)$ if
    $\displaystyle f(x)=\frac{k(x)}{x}$ if $\displaystyle x $ not $\displaystyle 0$
    $\displaystyle 0 $ if $\displaystyle x=0$

    and $\displaystyle k(0)=k'(0)=0$ and $\displaystyle k''(0)=17$

    My solution
    $\displaystyle f'(0)=lim_{h\rightarrow{0}} \frac{f(h)-f(0)}{h}$
    $\displaystyle =lim_{h\rightarrow{0}} \frac{\frac{k(h)}{h}}{h}$
    Question boils down to finding $\displaystyle lim_{h\rightarrow{0}} k(h)$

    But k is differentiable at 0, hence k is continuous at 0, so
    $\displaystyle lim_{h\rightarrow{0}} k(h)=0$
    Hence $\displaystyle f'(0)=0$
    END

    I am concerned about my solution because it doesn't use some of the information provided. Help please?
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  2. #2
    Senior Member
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    $\displaystyle \lim \frac{k(h)}{h^2} = \lim \frac{k'(h)}{2h} = \lim \frac{k''(h)}{2} = \frac{k''(0)}{2} = \frac{17}{2}$

    I applied L'Hopital's rule twice. All limits are taken as h goes to 0.
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