Product Rule Differentiation

Product rule differentiation

When a function is written as a product of two differentiable functions, differentiate it by using the product rule:

[ (uv)' = u'v + uv' ]

Here, choose one factor as (u) and the other as (v). Different choices are usually fine, as long as you stay consistent.

Method

1. Identify the two factors in the product.
2. Differentiate each factor separately to find (u') and (v').
3. Substitute into the product rule: (u'v + uv').
4. Simplify the result by combining like terms, expanding if needed, and writing the final answer neatly.

Example structure

If (f(x) = (x^2)(\sin x)), take (u = x^2) and (v = \sin x). Then [ f'(x) = 2x\sin x + x^2\cos x. ]

Check your work

A good check is to make sure both parts appear: one term should keep the first factor and differentiate the second, and the other should keep the second factor and differentiate the first. Also, confirm that the final expression is simplified as much as possible.