Derivative of arctan x

Differentiate again: \frac{d^2}{dx^2}\left[\arctan\left(x\right)\right] = -\frac{2 x}{\left(x^{2} + 1\right)^{2}}.

Plug into the derivative: f'(1) = 0.5. That is the slope of the tangent line there; drag the graph to see it.

\frac{d}{dx}\left[\arctan\left(x\right)\right] = \frac{1}{x^{2} + 1}

The inverse tangent flattens toward ±π/2, so its slope 1/(1 + x²) is largest (1) at the origin and fades toward 0 as |x| grows.

Inverse trig rule
$\frac{d}{dx}\left[\arctan\left(x\right)\right] = \frac{1}{x^{2} + 1}$

Drag the graph · at x = 1.20 the slope is 0.40984, the height of the dashed curve.

Questions students ask

What is the derivative of arctan x?

$\frac{d}{dx}\left[\arctan\left(x\right)\right] = \frac{1}{x^{2} + 1}$ .

What is the second derivative of arctan x?

Differentiate again: $\frac{d^2}{dx^2}\left[\arctan\left(x\right)\right] = -\frac{2 x}{\left(x^{2} + 1\right)^{2}}$ .

What is the slope of arctan x at x = 1?

Plug into the derivative: $f'(1) = 0.5$ . That is the slope of the tangent line there; drag the graph to see it.

How do you find the derivative of arctan x?

The inverse tangent flattens toward ±π/2, so its slope 1/(1 + x²) is largest (1) at the origin and fades toward 0 as |x| grows.

What is the integral of arctan x?

Use the integral calculator to find the antiderivative of arctan x step by step.