Question

find derivative by ab initio x+1/x-1

Answer 1

Let y=f(x)  = (x+1)/(x-1)

f(x+h) = (x+h+1)/(x+h-1)

Difference quotient = f(x+h)-f(x)/h

Numerator = (x+h+1)/(x+h-1)- (x+1)/(x-1)

={ (x+h+1)(x-1)-(x+h-1)(x+1)}/{(x+h-1)(x-1)}(1/h)$\frac{x^2-1+hx-h-(x^2-1+hx+h)}{(x+h-1)(x-1)h} </p><p>=\frac{-2h}{h(x+h-1)(x-1)}$

When x tends to 0, this becomes

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

=