Derivative of arctan(x^2): derivative with inverse tangent and chain rule.

We compute the derivative of arctan(x^2) by recognizing that this is a function composition, where x^2 is the inner function and arctan is the outer function. We need to start by taking the derivative of the inverse tangent with respect to the inner function, treating x^2 as a single variable for the derivative.

Comparing to the standard formula for the derivative of arctan(x), we simply evaluate the derivative at x^2 to obtain 1/(1+x^4). Next, the chain rule requires that we tack on the derivative of x^2 itself (the derivative of the inner function). This gives us a factor of 2x.

Finally, we simplify the derivative of arctan(x^2) to 2x/(1+x^4), and we're done!