Step by step
The question
Question: find the derivative of f(x) = (2x − 3)/(x² + 1).
1 Name u and v
Call the top u: 2x − 3.
Call the bottom v: x² + 1.
2 Find u′ and v′
The power rule on u gives u′ = 2.
On v it gives v′ = 2x.
3 Use the rule
The quotient rule: u′ v minus u v′, all over v squared.
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.
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) = (−2 x² + 6x + 2)/(x² + 1)².
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 2.
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 2.
Very close to 2.
Right!
Let's look back.
First: name u and v.
Then: find u′ and v′.
Then: use the rule.
Then: tidy the top.
Last: check.