Step by step
The question
Question: find the derivative of f(x) = (3x − 1)/(x + 2).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Name u and v
Call the top u: 3x − 1.
Call the bottom v: x + 2.
2 Find u′ and v′
The power rule on u gives u′ = 3.
On v it gives v′ = 1.
3 Use the rule
The quotient rule: u′ v minus u v′, all over v squared.
We check it with numbers at the end.
The order on top matters: u′ v comes first.
Careful!
Not u v′ minus u′ v!
Start with u′: the top's derivative comes first.
Put them in.
The bottom stays as v squared.
4 Tidy the top
What is in front of a bracket multiplies every part inside.
A minus in front of a bracket changes the sign of every part inside.
6 + 1 is 7.
3x − 3x is 0.
So f′(x) = 7/(x + 2)².
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 1.75.
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 1.75.
Very close to 1.75.
Right!
6 Pick the choice
Our answer is f′(x) = 7/(x + 2)².
That is choice D.
7 Why not the others
A is 3.
Wrong: divided the two derivatives.
B is −7/(x + 2)².
Wrong: took the top parts in the wrong order.
C is 5/(x + 2)².
Wrong: −(3x − 1) is −3x + 1: the minus goes to both.
Let's look back.
First: name u and v.
Then: find u′ and v′.
Then: use the rule.
Then: tidy the top.
Then: check.
Then: pick the choice.
Last: why not the others.