Differentiating Inverse Trigonometric Functions AB & BC

The six inverse trigonometric derivatives form three mirrored pairs. The radical family holds $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$, the rational family holds $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$, and the absolute-value family holds $\frac{d}{dx}\text{arcsec}\,x = \frac{1}{|x|\sqrt{x^2-1}}$. Each co-function derivative is the negative of its partner, so three forms plus one sign rule cover all six. Each comes from implicit differentiation plus a right triangle.

Two errors dominate. The first is crossing the families, pairing $\arctan$ with the radical or $\arcsin$ with the sum, or dropping the minus sign that distinguishes a co-function from its partner. The domains catch this quickly: $\arcsin$ is defined only on $[-1,1]$ and its derivative must blow up at the ends, while $\arctan$ is defined everywhere and its derivative must never be undefined. The second is treating the stated forms as complete when the argument is not a bare $x$, which drops the chain rule factor in the numerator and leaves the argument unsquared in the denominator.