Differentiating a Polynomial

Date: 04/25/99 at 02:56:24
From: Sarah Loader
Subject: Differentiation

Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4.

Date: 04/26/99 at 15:18:58
From: Doctor Jeff
Subject: Re: Differentiation

Hello, Sarah,

When differentiating a polynomial such as x^8 + 3x^5 + (3-5x)^4, you 
can differentiate each of the terms one at a time and then add them 
up to get the derivative of the whole thing.

Remember that the derivative of ax^n, where a and n are constants, is 
nax^(n-1). For example, the derivative of 

     10x^6 with respect to x is

     6*10x^(6-1) = 60x^5

Using this method takes care of the first two terms of your function. 
The third term requires the use of what is called the Chain Rule. 
Let's say you are trying to differentiate 

     (2x^2 + 3x + 5)^3

Think of (2x^2 + 3x + 5) as a function g(x). The chain rule states 
that the derivative of

     (g(x))^n with respect to x is

     n*(g(x))^(n-1)*g'(x)

     [Note that g'(x) is a common notation for the derivative of g(x)]

For the derivative of (2x^2 + 3x + 5)^3, we thus get

     3*(2x^2 + 3x + 5)^2*(4x + 3)

You can use this same method for differentiating the third term of 
the polynomial in your problem.

As a side note, you actually are using the Chain Rule every time you 
differentiate. If you are taking the derivative of x^n, you can 
consider x equal to the function g(x). Using the rule, you get

     der (g(x))^n 

     = n*(g(x))^(n-1)*g'(x)

But since g(x) = x and g'(x) = der x = 1, we get n*x^(n-1), which is 
the procedure you followed in differentiating the first two terms. 

Hope this helped, Sarah. Good luck with solving the problem. 

- Doctor Jeff, The Math Forum
  http://mathforum.org/dr.math/