Differentiate each of the following from first principle: `(2x+3)/(x-2)`

Explanation

`f'(x)=lim_(h->0)frac{f(x+h)-f(x)}{h}`

`=lim_(h->0)frac{frac{2(x+h)+3}{x+h-2}-frac{2x+3}{x-2}}{h}`

`=lim_(h->0)frac{(2x+2h+3)(x-2)-(x+h-2)(2x+3)}{h} xx lim_(h->0)frac{1}{(x-2)(x+h-2)}`

`=frac{1}{(x-2)^2}lim_(h->0)frac{2x^2-4x+2hx-4h+3x-6-2x^2-3x-2hx-3h+4x+6}{h}`

`=frac{1}{(x-2)^2} lim_(h->0) frac{-7h}{h}`

`=frac{1}{(x-2)^2} lim_(h->0) -7=>-frac{7}{(x-2)^2}`