Derivative of y = (x + 4)^5/(x + 1)^3 using logarithms Step by Step

Step by step

The question

Question: find dy/dx for y = (x + 4)⁵ over (x + 1)³, using logarithms.

y = (x + 4)⁵(x + 1)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 + 4)⁵(x + 1)³

2 Use the log rules

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

ln y = 5 ln(x + 4) − 3 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 = 5x + 4 − 3x + 1

4 Multiply by y

Multiply both sides by y.

dydx = y(5x + 4 − 3x + 1)

5 Put y back

Put y back.

dy/dx = (x + 4)⁵ over (x + 1)³ × (5/(x + 4) − 3/(x + 1)).

Books leave it like this.

dydx = (x + 4)⁵(x + 1)³(5x + 4 − 3x + 1)

6 Check

Let's check: the derivative is the slope.

At x = 2, our answer gives −48, with a calculator.

dy/dx at x = 2: −48

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

A calculator gives about −47.97.

Very close to −48.

Right!

y(2.001) − y(2)0.001 ≈ −47.97

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