Derivative of arctan(x) is 1/(1+x^2)

Answer

1(x2+1)\frac{1}{\left(x^{2}+1\right)}
(1+x^2)^(-1)

Derivative of atan(x)

Rule Used
Inverse Trigonometric Rule
Steps
  1. Recognize an inverse trigonometric function. It is not the reciprocal of the trig function: arcsin⁡(x)\arcsin(x) means arcsin⁡(x)\arcsin(x), the angle whose sine is x.
  2. The derivative comes from implicit differentiation of the inverse relation, then a Pythagorean identity that removes the angle.
  3. Final answer: (1+x2x^2)^(-1)

The derivative of arctan(x) is 1/(1 + x^2) for every real x.

Instant step-by-step solution for atan(x)

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Derivative of atan(x) = (1+x^2)^(-1) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-arctan-x

Formula for arctan(x), tan^-1(x), and atan(x)

d/dx arctan(x) = d/dx tan^-1(x) = d/dx atan(x) = 1 / (1 + x^2), for every real x.

Unlike arcsin and arccos, there is no square root and no gap in the domain. The derivative is always positive and it tends to 0 as x goes to either infinity, which matches the horizontal asymptotes of arctan. The calculator is filled with atan(x).

Open arctan(x) in the derivative calculator

From tan(y) = x to 1/(1+x^2)

Let y = arctan(x), so tan(y) = x and y lies in (-π/2, π/2). Differentiating both sides uses the known derivative of tangent: sec^2(y) · y' = 1.

Thus y' = 1/sec^2(y). The Pythagorean identity sec^2(y) = 1 + tan^2(y) rewrites the denominator without a trig function. Because tan(y) = x, the denominator is 1 + x^2.

The result 1/(1 + x^2) is defined for every real x. It is also the derivative of tan^-1(x) and of atan(x); those are names for the same function. It is not the derivative of tan(x), which is sec^2(x) and which is undefined at odd multiples of π/2.

  1. Set y = arctan(x), so tan(y) = x and y is in (-π/2, π/2).
  2. Differentiate both sides: sec^2(y) · dy/dx = 1.
  3. Solve: dy/dx = 1/sec^2(y).
  4. Use sec^2(y) = 1 + tan^2(y).
  5. Replace tan(y) with x: sec^2(y) = 1 + x^2.
  6. Conclude d/dx arctan(x) = 1/(1 + x^2), for every real x.

Worked examples, including d/dx atan(2x)

d/dx atan(x) at x = 1

  1. atan(x) is arctan(x). The derivative is 1/(1 + x^2).
  2. At x = 1 the denominator is 1 + 1 = 2.
  3. The slope is 1/2. Arctan(1) = π/4, and the tangent there is not horizontal.

Answer: 1/2

Solve this example

Derivative of arctan(2x)

  1. Outer derivative: 1/(1 + u^2) with u = 2x.
  2. Inner derivative: 2. The chain rule multiplies by 2 and squares the 2 inside.
  3. 1 + (2x)^2 = 1 + 4x^2. There is no domain restriction.

Answer: 2 / (1 + 4x^2)

Solve this example

Derivative of arctan(x) + arctan(1/x) for x > 0

  1. For x > 0 the sum arctan(x) + arctan(1/x) equals π/2, a constant.
  2. Differentiating confirms it: 1/(1 + x^2) plus the chain rule on arctan(1/x).
  3. The second term is [1/(1 + 1/x^2)] · (-1/x^2) = -1/(1 + x^2).
  4. The terms cancel. A nonzero answer means the derivative of 1/x was forgotten.

Answer: 0, for x > 0

Solve this example

Mistakes that swap arctan with tan

Answering sec^2(x)

sec^2(x) is the derivative of tan(x). The derivative of arctan(x) is the rational function 1/(1 + x^2). One is unbounded near π/2; the other never exceeds 1.

Writing 1/(1 - x^2)

The denominator is 1 plus x^2, because the identity is sec^2 = 1 + tan^2. A minus sign in the denominator is a different function (it shows up for artanh, not arctan) and it is undefined at x = ±1, where arctan is fine.

Chain rule on arctan(2x)

Two errors are common. Dropping the 2 gives 1/(1 + 4x^2). Forgetting to square the 2 gives 2/(1 + 2x^2). The derivative of the inside is 2, and (2x)^2 is 4x^2, so the answer is 2/(1 + 4x^2).

Restricting the domain to (-1, 1)

That restriction belongs to arcsin and arccos. Arctan accepts every real input, and 1 + x^2 is never zero, so the derivative has the same domain as the function.

Related functions

Frequently asked questions

What is the derivative of tan^-1(x)?

The derivative of tan^-1(x) is 1/(1 + x^2). tan^-1(x) means arctan(x), not 1/tan(x).

What is d/dx atan(x)?

atan(x) is the same function as arctan(x). d/dx atan(x) = 1/(1 + x^2), for every real x.

What is the derivative of arctan(2x)?

By the chain rule it is 2/(1 + (2x)^2), which simplifies to 2/(1 + 4x^2).

Is the derivative of arctan(x) equal to sec^2(x)?

No. sec^2(x) is the derivative of tan(x). The derivative of arctan(x) is 1/(1 + x^2).

Graph and rules

f(x) f'(x)

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Frequently Asked Questions

What is the derivative of atan(x)?▼
The derivative of atan(x) is (1+x^2)^(-1).
How do you find the derivative of atan(x)?▼
Apply the standard differentiation rules — the key rule used here is the Inverse Trigonometric Rule. The step-by-step result is (1+x^2)^(-1).
Why is the derivative of atan(x) equal to (1+x^2)^(-1)?▼
Because differentiation measures how a function changes — applying the rules to atan(x) yields (1+x^2)^(-1). See the full step-by-step breakdown above.

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