Question

help please ... nth derivative of ax+b/cx+d......how to decompose ax+b/cx+d into a/c+(bc-ad)/c(cx+d)....
alternative step : how to decompose ax+b/cx+d into (1/c)*[a(x+d/c)+(b-ad/c)]/(x+d/c)

Answer 1

Long or synthetic division would work.

which means
\[\frac{ax+b}{cx+d}=\frac{a}{c}+\frac{b-\frac{ad}{c}}{cx+d}\]
Multiplying the second term by \(\dfrac{c}{c}\) gives you the term you want.