# Derivative

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• Oct 17th 2010, 02:47 PM
wowo
Derivative
Hi.
How to find the derivative of 1/(x^2) using definition of the derivative;I know how to find it by using the power rule but not the definition(Happy)(Speechless).

Thanks.
• Oct 17th 2010, 02:51 PM
lvleph
The definition of the derivative is: If $\displaystyle f(x) = \frac{1}{x^2}$ then
$\displaystyle f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$
So try doing that and come back and tell us how you did. We can help you if you still need help.
• Oct 17th 2010, 03:06 PM
wowo
I need a hint, 'cause I'm still stuck.
• Oct 17th 2010, 03:07 PM
lvleph
Tell me what f(x+h) and f(x) are, then tell post what $\displaystyle \frac{f(x+h) - f(x)}{h}$. Once you have all that you should be able to simplify things and take the derivative.
• Oct 17th 2010, 03:09 PM
skeeter
$\displaystyle \displaystyle f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}$

start by getting a common denominator and get those two fractions in the numerator together.
• Oct 17th 2010, 03:35 PM
wowo
I keep getting lost during simplifying process.(Angry)(what's wrong with me(Headbang))
• Oct 17th 2010, 03:43 PM
skeeter
Quote:

Originally Posted by wowo
I keep getting lost during simplifying process. what's wrong with me

can' say ... you haven't shown anything other than posting the problem.

start by getting these two fractions together ...

$\displaystyle \displaystyle \frac{1}{(x+h)^2} - \frac{1}{x^2}$

... note that the common denominator is $\displaystyle x^2(x+h)^2$.

do it.
• Oct 17th 2010, 03:53 PM
wowo
$\displaystyle x^2-1(x^2+2xh+h^2)/(x^2+2xh+h^2)(x^2)/h$

Am I right????
• Oct 17th 2010, 03:55 PM
lvleph
Yes, continue simplification. Eventually an h will cancel.
• Oct 17th 2010, 03:55 PM
skeeter
Quote:

Originally Posted by wowo
$\displaystyle x^2-1(x^2+2xh+h^2)/(x^2+2xh+h^2)(x^2)/h$

Am I right????

keep going ... clean it up
• Oct 17th 2010, 04:26 PM
wowo
Okay, I'm completely lost with simplifying
• Oct 17th 2010, 04:32 PM
skeeter
Quote:

Originally Posted by wowo
Okay, I'm completely lost with simplifying

$\displaystyle \displaystyle \frac{1}{h} \left[\frac{x^2 - (x^2 + 2xh + h^2)}{x^2(x+h)^2}\right]$

distribute the negative in the numerator and combine like terms ... then factor out an $\displaystyle h$ from what's left in the numerator ...
• Oct 17th 2010, 04:58 PM
wowo
So, It's $\displaystyle 2x+h/(x^2(x+h)^2$
$\displaystyle 2x/(x^2)(x^2)$
$\displaystyle 2/x^3$
• Oct 17th 2010, 05:09 PM
mr fantastic
Quote:

Originally Posted by wowo
So, It's $\displaystyle 2x+h/(x^2(x+h)^2$
$\displaystyle 2x/(x^2)(x^2)$
$\displaystyle 2/x^3$

No. You have made a careless mistake in the numerator. Go back, check your work, find the mistake. Hint: You know what the final answer is meant to be, don't you ....?
• Oct 17th 2010, 05:22 PM
wowo
Is it -2x+h/(x^2(x+h)^2 ??
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