Question

Compute the derivative of $f\left(x\right)=\frac{x}{x-2}$ by
(a) Using the limit definition of the derivative
(b) Using the quotient rule

Answer 1

(a) By definition of limit

$f^{\prime }\left(x\right)=li{m}_{h->0}\frac{f(x+h)-f\left(x\right)}{h}$

$=li{m}_{h->0}\frac{x^2+hx-2x-2h-x^2-xh+2x}{(x+h-2)(x-2)h}$

$=li{m}_{h->0}\frac{-2h}{(x+h-2)(x-2)}$
$f^{\prime }\left(x\right)=\frac{-2}{(x-2{)}^2}$

(b) By quotient rule

$f^{\prime }\left(x\right)=\frac{1.(x-2)-1.\left(x\right)}{(x-2{)}^2}$

$f^{\prime }\left(x\right)=\frac{-2}{(x-2{)}^2}$