Derivative of arctan x: Formula, Proof and Examples

The derivative of \arctan x is \dfrac{1}{1 + x^2}. Unlike arcsin and arccos, there is no square root and no domain restriction: the formula works for every real number.

Graph of arctan x and its derivative

The derivative is a bell-shaped curve that is always positive, so \arctan x always increases. Because the derivative goes to 0 at both ends, the graph flattens out toward \pm\frac{\pi}{2}.

Proof of the derivative of arctan x

Let y = \arctan x, so \tan y = x with -\frac{\pi}{2} < y < \frac{\pi}{2}. Differentiate both sides with implicit differentiation, using the derivative of tan x:

The Pythagorean identity \sec^2 y = 1 + \tan^2 y turns this into x:

No sign choice is needed this time, since 1 + x^2 is always positive.

Chain rule version: derivative of arctan(u)

With the chain rule:

The last row gives the standard integral \int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C.

Worked examples

Example 1: arctan of 1/x. For x \ne 0,

This is the negative of the arctan derivative, so \arctan x + \arctan\frac{1}{x} is constant on each side of 0. It equals \frac{\pi}{2} for x > 0 and -\frac{\pi}{2} for x < 0.

Example 2: product rule. For f(x) = x\arctan x:

Example 3: tangent line. At x = 1, \arctan 1 = \frac{\pi}{4} and the slope is \frac{1}{2}. The tangent line is y = \frac{\pi}{4} + \frac{1}{2}(x - 1).

Common mistakes

1. Writing \sec^2 x. That is the derivative of \tan x, not of \arctan x.
2. Writing \frac{1}{1 + x}. The x is squared. Check: at x = 1 the slope must be \frac{1}{2}.
3. Forgetting to square the whole inside. \arctan(3x) gives 1 + 9x^2, not 1 + 3x^2.
4. Mixing it up with \frac{1}{\tan x} = \cot x. \tan^{-1} x is the inverse function.

Read backwards, this derivative gives the integral of 1/(1 + x²). The integral of arctan x itself uses integration by parts. See also the arctangent function and inverse trig derivative problems.