Calculus

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# Derivative of Arctan

#### Lesson Contents

## How do you Differentiate Arctangent?

The inverse tangent — known as arctangent or shorthand as arctan, is usually notated as tan^{-1}(some function). To differentiate it quickly, we have two options:**1.) Use the simple derivative rule.****2.) Derive the derivative rule, and then apply the rule.**

In this lesson, we show the derivative rule for tan^{-1}(u) and tan^{-1}(x). There are four example problems to help your understanding.

At the end of the lesson, we will see how the derivative rule is derived.

### Derivative of Arctan(u)

The derivative rule for arctan(u) is given as:

Where *u* is a function of a single variable, and the prime symbol * ‘* denotes the derivative with respect to that variable. Here are some examples of a single variable function

*u*:

u = x

u = sin(x)

u = y

^{3}– 3y + 4

### Derivative of Arctan(x)

The derivative rule for arctan(x) is the arctan(u) rule but with each instance of *u* replaced by *x*. Since the derivative of x is simply 1, the numerator simplifies to 1. The derivative rule for arctan(x) is given as:

Where * ‘* denotes the derivative with respect to x.

## Example Problems

### Derivative of Arctan(2x)

Find the derivative with respect to x of tan^{−1}(2x).

Solution:

### Derivative of Arctan(1/x)

Find the derivative with respect to x of tan^{−1}(1/x).

Solution:

### Derivative of Arctan(4x)

Find the derivative with respect to x of tan^{−1}(4x).

Solution:

### Derivative of Arctan(x^{2} + 1)

Find the derivative with respect to x of tan^{−1}(x^{2} + 1).

Solution:

## Bonus Lesson: What Makes Arctan Differentiable?

Arctan is a differentiable function because **its derivative exists on every point of its domain**. In the image below, a single period of arctan(x) is shown graphed. The curve is continuous and does not have any sharp corners.

If there is a sharp corner on a graph, the derivative is not defined at that point. So, if you come across a function whose graph has sharp corners, it will not be differentiable on every point of its domain.

The function f(x) = arctan(x) graphed for a single period.

### Proof of the Derivative Rule

Since arctangent means inverse tangent, we know that arctangent is the inverse function of tangent. **Therefore, we may prove the derivative of arctan(x) by relating it as an inverse function of tangent.** Here are the steps for deriving the arctan(x) derivative rule.

1.) y = arctan(x), so x = tan(y)

2.) dx/dy[x = tan(y)] = sec^{2}(y)

3.) Using sum of squares corollary: sec^{2}(y) = 1 + tan^{2}(y)

4.) tan^{2}(y) = x^{2} so dx/dy = 1 + x^{2}

5.) Flipping dx/dy, we get **dy/dx = 1/(1 + x ^{2})**