Step by step
The question
Question: find the derivative of f(x) = (x³ + 1)(2x − 1).
1 Name u and v
Call the first bracket u: x³ + 1.
Call the second bracket v: 2x − 1.
2 Find u′ and v′
The power rule on u gives u′ = 3 x².
On v it gives v′ = 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) = 8 x³ − 3 x² + 2.
5 Check
Let's check another way: multiply the brackets out first.
f(x) is 2 x⁴ − x³ + 2x − 1.
The power rule on that gives 8 x³ − 3 x² + 2.
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.