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### Differentiating a Polynomial

```
Date: 04/25/99 at 02:56:24
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

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/
```
Associated Topics:
High School Calculus

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