Derivative of atan(x) - Proof and Explanation

Proof

We want to find the derivative of arctan(x). Let y=arctan(x). Then, by definition, x=tan(y).

Taking the derivative of both sides with respect to x:

ddx(x)=ddx(tan(y)) 1=sec2(y)·dydx

Now, we solve for dydx:

dydx=1sec2(y)

Using the identity sec2(y)=1+tan2(y) and knowing tan(y)=x:

dydx=11+x2

Thus, the derivative of arctan(x) is:

ddxarctan(x)=11+x2

Explanation

To find the derivative of arctan(x), we start by letting y=arctan(x). This means that x=tan(y), representing the angle whose tangent is x.

Next, we differentiate both sides of the equation x=tan(y) with respect to x. The derivative of x with respect to x is simply 1.

On the right side, the derivative of tan(y) with respect to y is sec2(y), and by the chain rule, we multiply by dydx, which gives us sec2(y)·dydx.

Setting the derivatives equal, we get:

1=sec2(y)·dydx

We then solve for dydx by dividing both sides by sec2(y):

dydx=1sec2(y)

We use the trigonometric identity sec2(y)=1+tan2(y). Since tan(y)=x (from our earlier definition), we substitute x for tan(y):

sec2(y)=1+x2

Thus, the expression for the derivative simplifies to:

dydx=11+x2

Q.E.D.