Inverse trig derivatives

The inverse trigonometric functions — arcsin, arccos, arctan, and the inverses of the reciprocal functions, arccot, arcsec, and arccsc — have derivatives that are purely algebraic. The three most commonly used are d/dx arcsin(x) = 1/√(1 − x²), d/dx arccos(x) = −1/√(1 − x²), and d/dx arctan(x) = 1/(1 + x²). The remaining three follow the same pattern: d/dx arccot(x) = −1/(1 + x²), d/dx arcsec(x) = 1/(|x|√(x² − 1)), and d/dx arccsc(x) = −1/(|x|√(x² − 1)).

Two features make these easy to remember. Each "co-" function's derivative is simply the negative of its partner's, so learning three formulas gives you all six. And each derivative comes from implicit differentiation: writing y = arcsin(x) as sin(y) = x, differentiating both sides to get cos(y)·(dy/dx) = 1, and then replacing cos(y) with √(1 − x²) using the Pythagorean identity yields dy/dx = 1/√(1 − x²).

In practice these formulas are almost always paired with the chain rule, since problems rarely present a bare x inside the function. For an inner function u, d/dx arctan(u) = u′/(1 + u²). So the derivative of arctan(3x) is 3/(1 + 9x²), and the derivative of arcsin(x²) is 2x/√(1 − x⁴). Reading these formulas backwards also supplies several standard antiderivatives, which is why 1/(1 + x²) integrates to arctan(x) + C.

AP Calculus AB and BC both test inverse trig derivatives. Expect free-response and multiple-choice items that combine them with the chain rule, the product and quotient rules, or implicit differentiation, and recognize that the same formulas appear again in the integration unit as antiderivative patterns.