Question

F(x)= (x+1)/(x-1)

find the derivative using the definition

Answer 1

`\text{Given that,}\\\\f(x)= (x+1)/(x-1)\\\\\\f'(x)=\lim\limits_(h \to 0) (f(x+h) - f(x) )/(h)\\\\\\ ~~~~~~~~=\lim\limits_(h \to 0) \frac{\left( \tfrac{x+h+1}{x+h-1}\right) - \tfrac{x+1}{x-1}}{h}\\\\\\~~~~~~~~=\lim\limits_(h \to 0) \frac{\tfrac{(x+h+1)(x-1) - (x+1)(x+h-1)}{(x+h-1)(x-1)}}{h} \\\\\\~~~~~~~~=\lim\limits_(h \to 0) \frac{\tfrac{x^2-x+hx-h+x-1 - x^2-xh+x-x-h+1}{(x+h-1)(x-1)}}{h}\\\\\\~~~~~~~~`

`~~~~~~~=\lim\limits_(h \to 0) \frac{\tfrac{-2h}{(x+h-1)(x-1)}}{h}\\\\\\~~~~~~~~=\lim\limits_(h \to 0) (-2)/((x+h-1)(x-1))\\\\\\~~~~~~~~=(-2)/((x+0-1)(x-1))\\`

`\\\\~~~~~~~~=(-2)/((x-1)(x-1))\\\\\\~~~~~~~~=(-2)/((x-1)^2)`