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

Step by step

The question

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

y = (x + 1)²(x + 3)³(x + 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 + 1)²(x + 3)³(x + 2)

2 Use the log rules

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

ln y = 2 ln(x + 1) + 3 ln(x + 3) − ln(x + 2)

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 = 2x + 1 + 3x + 3 − 1x + 2

4 Multiply by y

Multiply both sides by y.

dydx = y(2x + 1 + 3x + 3 − 1x + 2)

5 Put y back

Put y back.

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

Books leave it like this.

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

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 285.9375, with a calculator.

dy/dx at x = 2: 285.9375

Now the slope between two very close points: x = 2 and x = 2.001.

A calculator gives about 286.04.

Very close to 285.9375.

Right!

y(2.001) − y(2)0.001 ≈ 286.04

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