# Differentiation of a polynomial function

• Duane
In summary, the conversation is about differentiating a rational function and the use of the quotient rule. The person asking the question mistakenly wrote the function as a polynomial and was advised to use the quotient rule instead of the chain rule. The notation of the function is corrected and a reminder to use LaTeX for mathematical symbols and equations is given.
Duane
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.

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 ?

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 $ax + \frac{b}{c} x + d$.

Better yet, look at the sticky thread:
Physics Forums > PF Lounge > Forum Feedback & Announcements
LaTeX Guide: Include mathematical symbols and equations in a post

## What is a polynomial function?

A polynomial function is a mathematical function that is made up of variables and coefficients, with each term consisting of a variable raised to a non-negative integer power. The highest power of the variable in the function is known as the degree of the polynomial.

## What is differentiation of a polynomial function?

Differentiation of a polynomial function is the process of finding the derivative of the function, which is the rate of change of the function with respect to its variable. This allows us to determine the slope of the function at any point and to analyze its behavior.

## Why is differentiation important in polynomial functions?

Differentiation is important in polynomial functions because it allows us to find the maximum and minimum values of the function, as well as the points where the function is increasing or decreasing. It also helps us to determine the concavity of the function, which is useful in solving optimization problems.

## What are the basic rules for differentiating a polynomial function?

The basic rules for differentiating a polynomial function are the power rule, the product rule, and the chain rule. The power rule states that the derivative of a variable raised to a power is equal to the power multiplied by the variable raised to the power minus one. The product rule is used when differentiating a product of two or more functions, and the chain rule is used when differentiating a composite function.

## How can differentiation be applied in real-life situations?

Differentiation can be applied in various real-life situations, such as in physics to determine the velocity and acceleration of a moving object, in economics to analyze the rate of change in demand and supply, and in engineering to optimize the design of structures and systems. It is also used in finance to calculate the rate of return on investments and in biology to study the growth and development of organisms.

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