Derivative of y = (x - 3)^3/(x + 1)^2 using logarithms Step by Step

Step by step

The question

Question: find dy/dx for y = (x − 3)³ over (x + 1)², using logarithms.

y = (x − 3)³(x + 1)²

The log rules turn a product into a sum, a fraction into a difference, and bring powers down.

ln(ab) = ln a + ln b

lnab = ln a − ln b

ln(a^n) = n ln a

Careful!

The ln of a product is a sum of lns, not a product.

Careful: not ln(ab) = ln a × ln b ✗

1 Take ln of both sides

Take ln of both sides.

ln y = ln(x − 3)³(x + 1)²

2 Use the log rules

Use the log rules: each power comes down, multiplying becomes adding, dividing becomes taking away.

ln y = 3 ln(x − 3) − 2 ln(x + 1)

3 The derivative of each side

The left side: 1/y, times dy/dx, by the chain rule.

Each ln(x + c) gives 1/x + c.

(ln y)′ = 1y × dydx

(ln(x + c))′ = 1x + c

Each power waits in front.

1ydydx = 3x − 3 − 2x + 1

4 Multiply by y

Multiply both sides by y.

dydx = y(3x − 3 − 2x + 1)

5 Put y back

Put y back.

dy/dx = (x − 3)³ over (x + 1)² × (3/(x − 3) − 2/(x + 1)).

Books leave it like this.

dydx = (x − 3)³(x + 1)²(3x − 3 − 2x + 1)

6 Check

Let's check: the derivative is the slope.

At x = 4, our answer gives 0.104, with a calculator.

dy/dx at x = 4: 0.104

Now the slope between two very close points: x = 4 and x = 4.001.

A calculator gives about 0.1.

Very close to 0.104.

Right!

y(4.001) − y(4)0.001 ≈ 0.1

Let's look back.

First: take ln of both sides.

Then: use the log rules.

Then: the derivative of each side.

Then: multiply by y.

Then: put y back.

Last: check.

Answer