Question

what is the second derivative of arctan x?

Answer 1

To find the second derivative of \( \arctan x \), we start with the first derivative.

1. The first derivative of \( \arctan x \) is given by:
\[
\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}
\]

2. Now, we need to find the second derivative, which is the derivative of the first derivative. We differentiate \( \frac{1}{1 + x^2} \):
\[
\frac{d}{dx}\left(\frac{1}{1 + x^2}\right)
\]

We can use the quotient rule or recognize that \( \frac{1}{1 + x^2} = (1 + x^2)^{-1} \) and use the chain rule. Let's differentiate using the chain rule:

\[
\frac{d}{dx}\left((1 + x^2)^{-1}\right) = -1 \cdot (1 + x^2)^{-2} \cdot \frac{d}{dx}(1 + x^2) = -1 \cdot (1 + x^2)^{-2} \cdot (2x)
\]

Simplifying this gives:
\[
-\frac{2x}{(1 + x^2)^2}
\]

So, the second derivative of \( \arctan x \) is:
\[
\frac{d^2}{dx^2}(\arctan x) = -\frac{2x}{(1 + x^2)^2}
\]