Differentiate the following function with respect to `x` from first principle: `(2x+3)/(3x+2)`

Explanation

Let `f(x)=frac{2x+3}{3x+2}` and we need to find its derivative i.e, `f'(x)`.

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

`f(x)=frac{2x+3}{3x+2}`

`f(x+h)=frac{2(x+h)+3}{3(x+h)+2}=frac{2x+2h+3}{3x+3h+2}`

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

`=lim_(h->0)frac{-5h}{h((3x+3h+2)(3x+2))}`

Putting `h=0` we get `=frac{-5}{(3x+3(0)+2)(3x+2)}`

`=frac{-5}{(3x+2)^2}`