# How to derive the derivative formula of arctan x?

1. Mar 29, 2007

### dnt

1. The problem statement, all variables and given/known data

basically what the topic states - derive the formula for the derivative of arctan x.

2. Relevant equations

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

3. The attempt at a solution

strange question because we already know the answer. but im not sure how to start this.

i know arctan x = y

therefore tan y = x

but what can i do with this? do i need to draw a right triangle and label all the sides? can someone help me get started? thanks.

2. Mar 29, 2007

### mattmns

Do you know the Inverse function theorem? Using it and a right triangle (as you said), gives the result pretty quickly.

3. Mar 29, 2007

### z-component

You could indeed draw a triangle knowing that the tangent is equal to x. In a triangle, if you're given that the tangent of some angle y is equal to x, what do you know about the lengths of the sides?

4. Mar 30, 2007

### dnt

the opposite would be x, the adjacent is 1 and the hypoteneuse is the square root of 1+x^2

after that im stuck.

5. Mar 30, 2007

### Galileo

You got tan y = x, you want dy/dx, so use implicit differentiation.

6. Mar 30, 2007

### Gib Z

$$y=arctan x$$

Therefore $$x=\tan y$$, Quite easy to see.

$$\frac{dx}{dy}=\sec^2 y$$

Using the Pythagorean Identity $\sec^2 y = \tan^2 y +1$ we can get this: $$\frac{dx}{dy}=\tan^2 y +1$$.

Flip the fraction since we want dy/dx, not dx/dy. And also, as seen on my second line tan y=x, so tan^2 y = x^2.

Thats how we get

$$\frac{dy}{dx}=\frac{1}{x^2+1}$$

7. Mar 30, 2007

### dextercioby

$$y=\arctan x$$

$$\tan y =x$$

$$\frac{d}{dx}\tan y= \frac{1}{\cos^{2}y}\frac{dy}{dx}=1$$

$$\frac{dy}{dx} =\cos^{2} y=\left(\frac{1}{\sqrt{1+x^{2}}}\right)^{2}=\frac{1}{1+x^{2}}$$