Finding derivatives

**I-Think** · Dec 6th 2010, 04:08 PM

Check my work please

Find $f'(0)$ if

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

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

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

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

I am concerned about my solution because it doesn't use some of the information provided. Help please?

**DrSteve** · Dec 6th 2010, 04:23 PM

$\lim \frac{k(h)}{h^2} = \lim \frac{k'(h)}{2h} = \lim \frac{k''(h)}{2} = \frac{k''(0)}{2} = \frac{17}{2}$

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

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