If f(x) = (x + 2)/(x - 1), what is f′(x)? Step by Step

Step by step

The question

Question: find the derivative of f(x) = (x + 2)/(x − 1).

f(x) = x + 2x − 1

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Name u and v

Call the top u: x + 2.

u = x + 2

Call the bottom v: x − 1.

v = x − 1

2 Find u′ and v′

The power rule on u gives u′ = 1.

u′ = 1

On v it gives v′ = 1.

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.

(uv)′ = u′v − uv′v²

Careful!

Not u v′ minus u′ v!

Start with u′: the top's derivative comes first.

Careful: not uv′ − u′v ✗

Put them in.

The bottom stays as v squared.

f′(x) = (1)(x − 1) − (x + 2)(1)(x − 1)²

4 Tidy the top

What is in front of a bracket multiplies every part inside.

= x − 1 − (x + 2)(x − 1)²

A minus in front of a bracket changes the sign of every part inside.

= x − 1 − x − 2(x − 1)²

−1 − 2 is −3.

x − x is 0.

So f′(x) = −3/(x − 1)².

= −3(x − 1)²

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives −3.

f′(0) = −3(−1)² = −3

Now the slope between two very close points: x = 0 and x = 0.001.

A calculator gives about −3.

Very close to −3.

Right!

f(0.001) − f(0)0.001 ≈ −3

6 Pick the choice

Our answer is f′(x) = −3/(x − 1)².

That is choice D.

D) −3x − 1^2

7 Why not the others

A is 1.

Wrong: divided the two derivatives.

A) 1 ✗ divided the two derivatives

B is 3/(x − 1)².

Wrong: took the top parts in the wrong order.

B) 3x − 1^2 ✗ took the top parts in the wrong order

C is 1/(x − 1)².

Wrong: −(x + 2) is −x − 2: the minus goes to both.

C) 1x − 1^2 ✗ −(x + 2) is −x − 2: 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.

Answer