# Thread: derivatives using limit formula.

1. ## derivatives using limit formula.

use limit formula to find

$\displaystyle f'(x) = \frac{1}{x^2}$

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

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

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

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

is this correct so far?

2. ## hi

Yes it is.
Now you can divide with h, so you´re left with:
$\displaystyle \frac{-2x-h}{x^2(x+h)^2}$

$\displaystyle \frac{-2x}{x^4 + 2(x^3)h +(x^2)(h^2)}$
This gives you, after you´ve let $\displaystyle h \, \rightarrow \, 0$
You also shorten out a x from the numerator and the divisor.
$\displaystyle \frac{-2}{x^3}$

3. Originally Posted by Twig
Yes it is.
Now you can divide with h, so you´re left with:
$\displaystyle \frac{-2x-h}{x^2(x+h)^2}$

$\displaystyle \frac{-2x}{x^4 + 2(x^3)h +(x^2)(h^2)}$
This gives you, after you´ve let $\displaystyle h \, \rightarrow \, 0$
You also shorten out a x from the numerator and the divisor.
$\displaystyle \frac{-2}{x^3}$
thanks mate!!!