Derivative of tan^-1(x)

Step by step

The question

Question: find the derivative of f(x) = tan⁻¹ x.

1 The rule

y = tan⁻¹ x means: tan y = x.

y is the angle whose tangent is x.

Take the derivative of both sides: sec² y times y′ is 1.

So y′ is 1 over sec² y, which is 1/1 plus tan² y: 1/1 plus x².

The rule: the derivative of tan⁻¹ x is 1/(1 + x²).

Careful!

The little −1 means the inverse function, the angle.

It does not mean 1 over tan x.

2 Use the rule

So f′(x) = 1/(1 + x²).

3 Check

Let's check: the derivative is the slope.

At x = 1, our answer gives 0.5, with a calculator in radians.

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

A calculator gives about 0.5.

Very close to 0.5.

Right!

Let's look back.

First: the rule.

Then: use the rule.

Last: check.