# Differentiation of products

1. Jan 1, 2005

### footprints

$$x^2(x+1)(x-2)^7$$
Could someone show me how to start?

2. Jan 1, 2005

### Benny

Well you currently have a product of three functions. I suggest starting by 'combining' the first two parts of the product as follows.

$$\frac{d}{{dx}}\left[ {x^2 \left( {x + 1} \right)\left( {x - 2} \right)^7 } \right] = \frac{d}{{dx}}\left[ {\left( {x^3 + x^2 } \right)\left( {x - 2} \right)^7 } \right]$$

Now just use the product rule to differentiate it.

3. Jan 1, 2005

### footprints

Ah!! I thought so to. But just wasn't sure. Thanks! Hapyy new year!

4. Jan 1, 2005

### dextercioby

In this simple case it works,as u're,mutiplying two simple polynomials.But what if thepolynomials had 50 terms?Would u do 2500 multiplications????
Here's the deal:the Leibniz rule is very general.It can be easily extended to finite arbitrary number of factors:
$$(ABC...Z)'=A'BC...Z+AB'C...Z+ABC'...Z+...+ABC...Z'$$
In your case,there are only 3 very simple polynomials.If u want to,u may not make the multiplications after the differentiation.

Daniel.

EDIT:'Prime' denotes differentiation.

Last edited: Jan 1, 2005
5. Jan 1, 2005

### footprints

Just checking, $$A'$$ is what I get after I differentiate A right?

Last edited: Jan 1, 2005
6. Jan 1, 2005

Yes it is.

7. Jan 1, 2005

### footprints

Thanks Nylex! Happy New Year!