# Derivative of Arctan – Formula & Examples

Derivative of Arctan: Before going to see what is the derivative of arctan, let us see some facts about arctan. Arctan (or) tan-1 is the inverse function of the tangent function. i.e., If y = tan-1x then tan y = x. Also, we know that if f and f-1 are inverse functions of each other then f(f-1(x)) = f-1(f(x)) = x. From this, tan(arctan x) = arctan(tan x) = x in the respective domains. We use these facts to find the derivative of arctan x.

Let us see the formula of derivative of arctan along with proof and a few solved examples.

## What Is Derivative of Arctan?

The derivative of arctan x is represented by d/dx(arctan x) (or) d/dx(tan-1x) (or) (arctan x)’ (or) (tan-1x)’. Its value is 1/(1+x2). We are going to prove it in two methods in the upcoming sections. The two methods are

• Using the chain rule
• Using the first principle

### Derivative of Arctan x Formula

The derivative of the arctangent function is,

• d/dx(arctan x) = 1/(1+x2) (OR)
• d/dx(tan-1x) = 1/(1+x2)

We are going to prove this formula now in the next sections.

## Derivative of Arctan Proof by Chain Rule

We find the derivative of arctan using the chain rule. For this, assume that y = arctan x. Taking tan on both sides,

tan y = tan (arctan x)

By the definition of inverse function, tan (arctan x) = x. So the above equation becomes,

tan y = x … (1)

Differentiating both sides with respect to x,

d/dx (tan y) = d/dx(x)

We have d/dx (tan x) = sec2x. Also, by chain rule,

sec2y · dy/dx = 1

dy/dx = 1/sec2y

Using one of the trigonometric identities, sec2y = 1 + tan2y.

dy/dx = 1/(1 + tan2y)

dy/dx = 1/(1 + x2) (from (1))

Substituting y = arctan x back here,

d/dx (arctan x) = 1/(1 + x2)

Hence proved.

## Derivative of Arctan Proof by First Principle

The derivative of a function f(x) by the first principle is given by the limit, f'(x) = limₕ→₀ [f(x + h) – f(x)] / h. To find the derivative of arctan x, assume that f(x) = arctan x. Then f(x + h) = arctan (x + h). Substituting these values in the above limit,

f'(x) = limₕ→₀ [arctan (x + h) – arctan x] / h

By inverse trigonometric formulas, We have arctan x – arctan y = arctan [(x – y)/(1 + xy)]. Apply this, we get

f'(x) = limₕ→₀ [arctan[(x + h – x)/(1 + (x + h) x)] ] / h

= limₕ→₀ (1/h) [arctan [ h / (1 + x2 + hx) ] ]

We have arctan x = x – x3/3 + x5/5 – … Using this, we get

f'(x) = limₕ→₀ (1/h) [ h / (1 + x2 + hx) – [h / (1 + x2 + hx)]3 / 3 + [h / (1 + x2 + hx)]/ 5 – ….]

= limₕ→₀ (1/h) [ h / (1 + x2 + hx) – [h3 / [3(1 + x2 + hx)3] + [h5 / [5(1 + x2 + hx)5]- ….]

Distributing (1/h),

f'(x) = limₕ→₀ [ 1 / (1 + x2 + hx) – [h2 / [3(1 + x2 + hx)3] + [h4 / [5(1 + x2 + hx)5]- ….]

Applying the limit h→0,

f'(x) = 1 / (1 + x+ 0) – 0 + 0 – …

f'(x) = 1/(1 + x2)

Hence proved.

Topics Related to Derivative of Arctan:

Here are some topics that are related to the derivative of arctan x.

## FAQs

What is Derivative of Arctan?

The derivative of arctan x is 1/(1+x2). i.e., d/dx(arctan x) = 1/(1+x2). This also can be written as d/dx(tan-1x) = 1/(1+x2).

How to Prove Derivative of Arctan Formula?

To derive the derivative of arctan, assume that y = arctan x then tan y = x. Differentiating both sides with respect to y, then sec2y = dx/dy. Taking reciprocal on both sides, dy/dx = 1/(sec2y) = 1/(1+tan2y) = 1/(1+x2).

What is the Derivative of Arctan x/2?

We have the derivative of arctan x to be 1/(1 + x2). By using this and chain rule, the d/dx(arctan x/2) = 1/(1+(x/2)2) d/dx (x/2) = 1/(1 + (x2/4)) · (1/2) = [4/(4 + x2)] · (1/2) = 2/(4 + x2).

What is the Derivative of Arctan √x?

We know that the derivative of arctan x to be 1/(1 + x2). By using this formula and chain rule, the d/dx(arctan √x) = 1/(1+(√x)2) d/dx (√x) = 1/(1 + x) · (1/2√x) = 1/(2√x(1+x)).

Is Arctan the Derivative of Tan?

No, the derivative of arctan is NOT tan. The derivative of arctan x is 1/(1 + x2).

What is the Difference Between the Derivatives of Tan x and Arctan x?

The derivative of tan x is sec2x whereas the derivative of arctan x is 1/(1+x2).

What is the Derivative of Arctan x/a?

The derivative of arctan x to be 1/(1 + x2). By using this and chain rule, the d/dx(arctan x/a) = 1/(1+(x/a)2) d/dx (x/a) = 1/(1 + (x2/a2)) · (1/a) = [a2/(a2 + x2)] · (1/a) = a/(a2 + x2).