Step by step
The question
Question: find the derivative of f(x) = (3x − 2)(x² + 2x).
1 Name u and v
Call the first bracket u: 3x − 2.
Call the second bracket v: x² + 2x.
2 Find u′ and v′
The power rule on u gives u′ = 3.
On v it gives v′ = 2x + 2.
3 Use the rule
The product rule: the derivative of u times v is u′ v plus u v′.
Put them in.
Careful!
Not u′ times v′!
Each part needs the other bracket.
4 Open the brackets
What is in front of a bracket multiplies every part inside.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
So f′(x) = 9 x² + 8x − 4.
5 Check
Let's check another way: multiply the brackets out first.
f(x) is 3 x³ + 4 x² − 4x.
The power rule on that gives 9 x² + 8x − 4.
The same!
Right!
Let's look back.
First: name u and v.
Then: find u′ and v′.
Then: use the rule.
Then: open the brackets.
Last: check.