Derivative of ln(x^2 + 1)

Step by step

The question

Question: find dy/dx for y = ln(x² + 1).

1 The inside

There is something inside: x² + 1.

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 x² + 1 gives 2x.

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, 2x.

So dy/dx = 2x/(x² + 1).

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.