Finding the Derivative...

• March 2nd 2010, 08:11 AM
KarlosK
Finding the Derivative...
We are just being introduced the the derivative concept and I was wondering if I understand this correctly.

We got the formula f(x) lim h->0 (f(x+h) - f(x)) / h

One example says: Find derivative for the function f(x)=x/(x-1)

Does this like correct:

1. (x/x-1+h) - (x/(x-1) / h
The x/x-1 - x/x-1 cancel out
Leaving you with H/H
so the derivative is 1??
• March 2nd 2010, 08:39 AM
nehme007
If $f(x) = \frac{x}{x-1}$, then $f(x+h) = \frac{x+h}{x+h-1}$ (you have to replace every x with x+h). Then...

$f(x+h) - f(x) = \frac{x+h}{x+h-1} - \frac{x}{x-1} = \frac{(x+h)(x-1)-x(x+h-1)}{(x+h-1)(x-1)}$

$=\frac{x(x-1)+hx-h-x(x-1)-hx}{(x+h-1)(x-1)}=\frac{-h}{(x+h-1)(x-1)}$

Dividing by h and taking the limit as h goes to 0, you get $\frac{-1}{(x-1)^2}$.