What is the second derivative of inverse tangent?

The second derivative of the inverse tangent function (also known as the arc tangent or arctan function) can be calculated by differentiating the first derivative.

The first derivative of the inverse tangent function, denoted as atan(x) or arctan(x), is given by:

d/dx [atan(x)] = 1 / (1 + x^2)

To find the second derivative, we differentiate the first derivative with respect to x:

d^2/dx^2 [atan(x)] = d/dx [1 / (1 + x^2)]

To simplify the calculation, we can rewrite the function as:

f(x) = (1 + x^2)^(-1)

Now, using the quotient rule, we can differentiate f(x) with respect to x:

f'(x) = -2x / (1 + x^2)^2

Next, we differentiate f'(x) using the quotient rule again:

f''(x) = [(-2)(1 + x^2)^2 - (-2x)(2)(1 + x^2)(2x)] / (1 + x^2)^4

Simplifying this expression further, we get:

f''(x) = [(-2)(1 + x^2) - (-4x^2)(1 + x^2)] / (1 + x^2)^4
= [-2 - 2x^2 + 4x^2(1 + x^2)] / (1 + x^2)^4
= [-2 - 2x^2 + 4x^2 + 4x^4] / (1 + x^2)^4
= (2 + 2x^2 + 4x^4) / (1 + x^2)^4

Therefore, the second derivative of the inverse tangent function, atan(x) or arctan(x), is given by:

d^2/dx^2 [atan(x)] = (2 + 2x^2 + 4x^4) / (1 + x^2)^4