**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^{-1}x 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^{-1}x) (or) (arctan x)’ (or) (tan^{-1}x)’. Its value is 1/(1+x^{2}). 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+x**^{2}) (OR)**d/dx(tan**^{-1}x) = 1/(1+x^{2})

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) = sec^{2}x. Also, by chain rule,

sec^{2}y · dy/dx = 1

dy/dx = 1/sec^{2}y

Using one of the trigonometric identities, sec^{2}y = 1 + tan^{2}y.

dy/dx = 1/(1 + tan^{2}y)

dy/dx = 1/(1 + x^{2}) (from (1))

Substituting y = arctan x back here,

d/dx (arctan x) = 1/(1 + x^{2})

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 + x^{2} + hx) ] ]

We have arctan x = x – x^{3}/3 + x^{5}/5 – … Using this, we get

f'(x) = limₕ→₀ (1/h) [ h / (1 + x^{2} + hx) – [h / (1 + x^{2} + hx)]^{3} / 3 + [h / (1 + x^{2} + hx)]^{5 }/ 5 – ….]

= limₕ→₀ (1/h) [ h / (1 + x^{2} + hx) – [h^{3} / [3(1 + x^{2} + hx)^{3}] + [h^{5} / [5(1 + x^{2} + hx)^{5}]- ….]

Distributing (1/h),

f'(x) = limₕ→₀ [ 1 / (1 + x^{2} + hx) – [h^{2} / [3(1 + x^{2} + hx)^{3}] + [h^{4} / [5(1 + x^{2} + hx)^{5}]- ….]

Applying the limit h→0,

f'(x) = 1 / (1 + x^{2 }+ 0) – 0 + 0 – …

f'(x) = 1/(1 + x^{2})

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+x^{2}). i.e., d/dx(arctan x) = 1/(1+x^{2}). This also can be written as d/dx(tan^{-1}x) = 1/(1+x^{2}).

**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 sec^{2}y = dx/dy. Taking reciprocal on both sides, dy/dx = 1/(sec^{2}y) = 1/(1+tan^{2}y) = 1/(1+x^{2}).

**What is the Derivative of Arctan x/2?**

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

**What is the Derivative of Arctan √x?**

We know that the derivative of arctan x to be 1/(1 + x^{2}). 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 + x^{2}).

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

The derivative of tan x is sec^{2}x whereas the derivative of arctan x is 1/(1+x^{2}).

**What is the Derivative of Arctan x/a?**

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