Quotient rule of derivative is: $(\frac{f}{g})^{\prime}$ = $(\frac{f^{\prime}g - g^{\prime}f}{g^2})$ but when I compute a deriative of $\frac{1}{(1-x)}$ , it gives $\frac{1}{(1-x)^2}$ which is right but taking second derivative of this gives me $\frac{2}{(1-x)^4}$ but the right answer is $\frac{2}{(1-x)^3}$. Where do I make a mistake?
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$\begingroup$ How do you get $\frac{2}{(1-x)^{4}}$? If we take $f=1$ and $g=(1-x)^{2}$. Than as in previous $f'=0$, $g'=-2(1-x)$. So you'll have $\frac{-(1\cdot (-2) \cdot(1-x))}{(1-x)^{4}}$. Which will yield $\frac{2}{(1-x)^{3}}$. $\endgroup$– kolobokishJan 21, 2017 at 12:35
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$\begingroup$ i was computing Taylor series and i see that i filled directly value of x = 0 , which was a mistake, :( thanks for the answer :-) $\endgroup$– Khan SaabJan 21, 2017 at 12:37
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$\begingroup$ Oh. Then I'm sorry.) $\endgroup$– kolobokishJan 21, 2017 at 12:39
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$\begingroup$ I think that using the formula you mentioned to compute the successive derivatives of $x\mapsto\frac{1}{1-x}$ is not the best point of view. It's easier if you consider using $\frac{d}{dx}\left(\frac{1}{g(x)}\right)=-\frac{g'(x)}{g(x)^2}$ $\endgroup$– AdrenJan 21, 2017 at 12:45
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$\begingroup$ When asking for someone to point out where you made a mistake, show your work—we’re not all mind-readers. $\endgroup$– amdJan 21, 2017 at 22:01
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3 Answers
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$\require{cancel}$ $\Big(\dfrac{1}{(1-z)^2}\Big)'=\dfrac{0\times(1-z)^2-2(-1)(1-z)\times1}{(1-z)^4}=\dfrac{2(1-z)}{(1-z)^4}=\dfrac{2\cancel{(1-z)}}{(1-z)^\cancel{4}}=\dfrac{2}{(1-z)^3}$
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$\begingroup$ It's a prime and not a second derivative of $1/(1-x)^2$ $\endgroup$– marwalixJan 21, 2017 at 12:47
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Let's derive
$$\left({1\over (1-x)^2}\right)^{'}={(1)'\cdot (1-x)^2-1\cdot\left((1-x)^2\right)^{'}\over (1-x)^4}$$
Now $(1)'=0$, $\left((1-x)^2\right)^{'}=-2(1-x)$ and so the derivative rewrites as
$$\left({1\over (1-x)^2}\right)^{'}={2(1-x)\over (1-x)^4}={2\over (1-x)^3}$$
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You get this because you can cancel out one $(1-x)$ in the denominator.
$\Big(\frac{1}{(1-x)^2}\Big)'=\frac{0*((1-x)^2)'-1*((1-x)^2)'}{((1-x)^2)^2}=\frac{2(1-x)}{(1-x)^4}=\frac{2}{(1-x)^3}$.
If you just follow the formula you had carefully, with $f=1$ and $g=(1-x)^2$, this is what you get.
answered Jan 21, 2017 at 12:39