Differentiate the following function with respect of `x :(a x+b)//(c x+d)`

Explanation

Let `f(x) = (ax + d)/(cx+d)`
`f(x+h) = (a(x+h)+b)/(c(x+h)+d)`
`therefore d/(dx)f(x)= underset(hto0)"lim"1/h[f(x+h)-f(x)]`
`=underset(hto0)"lim"1/h[(a(x+h)+b)/(c(x+h)+d)-(ax+b)/(cx+d)]`
`=underset(hto0)"lim"1/h[(ax+b+ah)/(c(x+h)+d)-(ax+b)/(cx+d)]`
`=underset(hto0)"lim"1/h[((ax+ah+b)(cx+d)-(ax+b)(cx+ch+d))/((c(x+h)+d)(cx+d))]`
`=underset(hto0)"lim"1/h[(acx^(2)+achx+bcx+adx+adh+bd-acx^(2)+achx+adx+bcx+bch+bd)/(c(x+h)+d(cx+d))]`
`underset(hto0)"lim"1/h[(acx^(2)+achx+bcx+adx+adh+bd-acx^(2)-achx-adx-bcx-bch-bd)/(c(x+h)+d(cx+d))]`
`underset(hto0)"lim"1/h[(adh-bch)/(c(x+h)+d(cx+d)]]`
`=(ac-bd)/(c(x+h)+d(cx=d))`
`=(ac-bd)/(cx+d)^(2)`