Derivative of x/(x + 1) Step by Step | Product and quotient rules

Step by step

The question

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

f(x) = xx + 1

1 Name u and v

Call the top u: x.

u = x

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.

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)(1)(x + 1)²

4 Tidy the top

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

= x + 1 − (x)(x + 1)²

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

= x + 1 − x(x + 1)²

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) = 1/(x + 1)².

= 1(x + 1)²

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives 1.

f′(0) = 1(1)² = 1

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

A calculator gives about 1.

Very close to 1.

Right!

f(0.001) − f(0)0.001 ≈ 1

Let's look back.

First: name u and v.

Then: find u′ and v′.

Then: use the rule.

Then: tidy the top.

Last: check.

Answer