Calculus BC · Unit 3: Differentiation: Composite, Implicit, and Inverse Functions · 14 min read · Updated 2026-05-11
Differentiating inverse trigonometric functions — AP Calculus BC
AP Calculus BC · Unit 3: Differentiation: Composite, Implicit, and Inverse Functions · 14 min read
1. Deriving Derivatives of Basic Inverse Trigonometric Functions★★☆☆☆⏱ 4 min
All derivative formulas for inverse trigonometric functions can be derived from the definition of inverse functions and implicit differentiation. The process follows the same steps for all six functions: rewrite the inverse trig equation as a regular trig equation, differentiate implicitly, solve for $\frac{dy}{dx}$, use a Pythagorean identity to rewrite in terms of $x$, and adjust the sign to match the principal range of the inverse function.
2. Differentiating Composite Inverse Trigonometric Functions★★★☆☆⏱ 4 min
Virtually all AP exam problems on this topic involve composite inverse trigonometric functions, where the argument of the inverse trig function is another function of $x$. To differentiate these, you must combine the inverse trig derivative rule with the chain rule: if $y = f(u(x))$ where $f$ is an inverse trig function, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
3. Differentiating Combinations of Inverse Trigonometric Functions★★★☆☆⏱ 3 min
AP problems often require differentiating combinations of inverse trig functions with other function types (polynomials, exponentials, logarithms) using the product rule or quotient rule. The process is identical to differentiating any other combination: apply the product or quotient rule first, then compute the derivative of each inverse trig term using the inverse trig rule and chain rule.
4. AP-Style Practice★★★★☆⏱ 3 min
Common Pitfalls
Why: Students remember the basic derivative of $\arcsin x$ but forget the inner function $5x$ has a non-zero derivative, incorrectly assuming $u' = 1$.
Why: Students mix up the derivatives of $\arcsin x$ and $\arccos x$, which only differ by a sign.
Why: Students forget the derivative of $\text{arcsec}$ requires an absolute value from domain restrictions.
Why: Students rush and only substitute the inner function into the chain rule term, not into the inverse trig derivative formula.
Why: Students assume $\arcsin(\sin f(x)) = f(x)$ for all $x$, which is only true when $f(x)$ falls in the principal range of $\arcsin$.