# Math Help - derivative

1. ## derivative

find the derivative of $y=\frac{x+1}{x-1}$ using first principles

so

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

$
\frac{(x+h)+1}{(x+h)-1} - \frac{x+1}{x-1}$
all over h

2. Originally Posted by euclid2
find the derivative of $y=\frac{x+1}{x-1}$ using first principles

so

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

$
\frac{(x+h)+1}{(x+h)-1} - \frac{x+1}{x-1}$
all over h

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

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

$=\frac{\color{blue}(x-1)(x+1)\color{black}+(x-1)h\color{blue}-(x+1)(x-1)\color{black}-(x+1)h}{(x-1)(x-1)+h(x-1)}$

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

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

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

setting h=0, the derivative is

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