# Differentiation of a polynomial function

## Main Question or Discussion Point

Hi guys

I'm getting into a little trouble when differentiating polynomial functions.

How do you differentiate

f(x)=ax+b/cx+d ?

Is there other ways of calculating this apart from the chain rule ?

Thanks for any help.

HallsofIvy
Homework Helper
Surely by the time you are taking Calculus you should know that what you give is NOT a polynomial!! And I can't imagine why you would think about the chain rule. That is a quotient so use the quotient rule.

Sorry I messed up.
What I meant was a "rational" function.

Are "rational" and "quotient" synonymous ?

By applying the quotient rule, I get

(ax+b/cx+d)'= [(cx+d)(ax+b)'-(ax+b)(cx+d)'] / (cx+d)^2

= [a(cx+d)-c(ax+b)] / (cx+d)^2

how to proceed ?

Stephen Tashi
You can multiply the terms in the top of the fraction $\frac{a(cx+d) - c(ax+b)}{(cx+d)^2}$ and then combine like terms. I don't see anything that "simplifies" beyond that.
A point about notation, you should write the original function as (ax+b)/(cx+d) to mean $\frac{ax+b}{cx+d}$ instead of writing it as (ax +b/cx + d}, which means $ax + \frac{b}{c} x + d$.