Derivative of y = (x - 1)(x + 4)^2/(x + 2)^3 using logarithms Step by Step

Step by step

The question

Question: find dy/dx for y = (x − 1)(x + 4)² over (x + 2)³, using logarithms.

y = (x − 1)(x + 4)²(x + 2)³

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)(x + 4)²(x + 2)³

2 Use the log rules

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

ln y = ln(x − 1) + 2 ln(x + 4) − 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 = 1x − 1 + 2x + 4 − 3x + 2

4 Multiply by y

Multiply both sides by y.

dydx = y(1x − 1 + 2x + 4 − 3x + 2)

5 Put y back

Put y back.

dy/dx = (x − 1)(x + 4)² over (x + 2)³ × (1/(x − 1) + 2/(x + 4) − 3/(x + 2)).

Books leave it like this.

dydx = (x − 1)(x + 4)²(x + 2)³(1x − 1 + 2x + 4 − 3x + 2)

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives 0.3281, with a calculator.

dy/dx at x = 2: 0.3281

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

A calculator gives about 0.33.

Very close to 0.3281.

Right!

y(2.001) − y(2)0.001 ≈ 0.33

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