Step by step
The question
Question: find the derivative of f(x) = (x² + 1)(3x − 2).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Name u and v
Call the first bracket u: x² + 1.
Call the second bracket v: 3x − 2.
2 Find u′ and v′
The power rule on u gives u′ = 2x.
On v it gives v′ = 3.
3 Use the rule
The product rule: the derivative of u times v is u′ v plus u v′.
Each bracket takes its turn to change, while the other one stays.
At the end we check it by opening the brackets first.
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.
6x² + 3x² is 9x².
So f′(x) = 9 x² − 4x + 3.
5 Check
Let's check another way: multiply the brackets out first.
f(x) is 3 x³ − 2 x² + 3x − 2.
The power rule on that gives 9 x² − 4x + 3.
The same!
Right!
6 Pick the choice
Our answer is f′(x) = 9x² − 4x + 3.
That is choice B.
7 Why not the others
A is 6x.
Wrong: multiplied the two derivatives.
That is the Careful mistake from before.
C is 6x² − 4x + 3.
Wrong: lost the 3x² from 3(x² + 1).
D is 9x² + 3.
Wrong: lost the −4x from 2x(3x − 2).
Let's look back.
First: name u and v.
Then: find u′ and v′.
Then: use the rule.
Then: open the brackets.
Then: check.
Then: pick the choice.
Last: why not the others.