1. 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.

Thanks.

2. 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.

3. I need a hint, 'cause I'm still stuck.

4. 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.

5. $\displaystyle \displaystyle f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}$

start by getting a common denominator and get those two fractions in the numerator together.

6. I keep getting lost during simplifying process.(what's wrong with me)

7. 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.

8. $\displaystyle x^2-1(x^2+2xh+h^2)/(x^2+2xh+h^2)(x^2)/h$

Am I right????

9. Yes, continue simplification. Eventually an h will cancel.

10. 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

11. Okay, I'm completely lost with simplifying

12. 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 ...

13. So, It's $\displaystyle 2x+h/(x^2(x+h)^2$
$\displaystyle 2x/(x^2)(x^2)$
$\displaystyle 2/x^3$

14. 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 ....?

15. Is it -2x+h/(x^2(x+h)^2 ??

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