Derivative of ln(3x)

Step by step

The question

Question: find dy/dx for y = ln(3x).

1 The inside

There is something inside: 3x.

Call it u.

Then y is just ln u.

2 The outside

The derivative of ln x is 1 over x.

First the outside.

The rule on ln u gives 1 over u.

3 The inside's slope

Now the inside.

The power rule on 3x gives 3.

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

Put the inside back in place of u.

Careful!

Do not stop here!

We must multiply by the inside's slope, 3.

Multiply: 3/(3x).

Divide the top and the bottom by 3.

So dy/dx = 1 over x.

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 1, with a calculator.

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 1.

Very close to 1.

Right!

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.