Derivative of $\arctan\left(x\right)$

Q: What is the derivative of atan(x)?

The derivative of ≤ft(x) is 11 + x^2.

The derivative of $\arctan\left(x\right)$ is $\frac{1}{1 + x^{2}}$ .

f(x) =

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

Type any function of x; the derivative is computed symbolically in your browser.

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

Explain it step by step in chat Make a short video of it

Graph of f(x) = atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.5.

How to work it out

1
We differentiate $f(x) = \arctan\left(x\right)$ with respect to x. The rule that applies is the standard derivative.
$f(x) = \arctan\left(x\right)$
2
This is a standard derivative from the table.
3
Simplify:
$f'(x) = \frac{1}{1 + x^{2}}$
4
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}$

LaTeX source of the formulas on this page

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

Values of f and f' for atan(x)

x	f(x)	f'(x)
-2	-1.1071487	0.2
-1	-0.78539816	0.5
0.5	0.46364761	0.8
1	0.78539816	0.5
2	1.1071487	0.2
3	1.2490458	0.1

Questions

What is the derivative of atan(x)?

The derivative of $\arctan\left(x\right)$ is $\frac{1}{1 + x^{2}}$ .

What is the second derivative of atan(x)?

-(2x) / (1 + $x^{2}$ )^2.

Which rule do you use for atan(x)?

The standard derivative.

What is the slope of atan(x) at x = 1?

0.5.

Derivative of arctan⁡(x)\arctan\left(x\right)arctan(x)

How to work it out

Values of f and f' for atan(x)

Questions

Related derivatives

Derivative of $\arctan\left(x\right)$