On-line Math 21

2.2  Differentiation Rules

Example 5 Find the derivative of

f(x) = (x+2)(x2+1) (x+3)(x2-2) .

Solution

This is more complicated, since you have products in the numerator and denominator, as well as the quotient. But the thing to look at is that the last thing you do in constructing this function is to take the quotient, so the first thing you need to handle in taking the derivative is the quotient. You always work from the outside in.

The first step, then, is to apply the quotient rule:

f¢(x) = æ ç è (x+2)(x2+1) (x+3)(x2-2) ö ÷ ø ¢ = ( (x+2)(x2+1)) ¢(x+3)(x2-2)-( (x+3)(x2-2)) ¢(x+2)(x2+1) (x+3)(x2-2) .

Next step

Then, apply the product rule twice in the numerator. Rather than write out all the steps of the product rule, you should be able to see how it is applied here directly.

f¢(x) = æ ç è (x+2)(x2+1) (x+3)(x2-2) ö ÷ ø ¢ = ( (x+2)(x2+1)) ¢(x+3)(x2-2)-( (x+3)(x2-2)) ¢(x+2)(x2+1) (x+3)(x2-2) = ( 1(x2+1)+(x+2)(2x)) (x+3)(x2-2)-( 1(x2-2)+(x+3)(2x)) (x+2)(x2+1) (x+3)(x2-2) .

As before, this is where you should stop. Yes, it can be simplified somewhat, but we don't do that until we need to.

Copyright (c) 2000 by David L. Johnson.