Topic 3.4 Differentiating Inverse Trigonometric Functions: state and apply the derivatives of the inverse trigonometric functions.

What this topic is asking

The College Board (Topic 3.4) wants you to know the derivatives of the inverse trigonometric functions - $\arcsin x$, $\arccos x$, $\arctan x$ and the rest - and to apply them, usually together with the chain rule. The surprising feature is that these derivatives are algebraic (rational and radical expressions in $x$) even though the functions themselves are transcendental.

The derivatives

For AP Calculus AB, the indispensable three are $\arcsin$, $\arccos$ and $\arctan$; the others appear less often but follow the same negative-partner pattern.

The pattern that cuts the memory load

Where these come from

Each derivative is obtained by implicit differentiation of the defining relationship - this is exactly the inverse-function idea from Topic 3.3 applied to trig.

Combining with the chain rule

In practice almost every exam appearance has a composite argument, so the chain rule rides along: $\frac{d}{dx}[\arcsin(u)] = \frac{u'}{\sqrt{1 - u^2}}$, $\frac{d}{dx}[\arctan(u)] = \frac{u'}{1 + u^2}$, and so on, where $u$ is whatever sits inside. For $\arctan(3x)$ the inner derivative is $3$, giving $\frac{3}{1 + 9x^2}$; for $\arcsin(x^2)$ the inner derivative is $2x$, giving $\frac{2x}{\sqrt{1 - x^4}}$. The pattern is identical to every other chain-rule problem: write the outer derivative with the inner copied in, then multiply by the inner derivative. Because the answers are purely algebraic, these derivatives are also the antiderivatives you will recognize in Unit 6 - seeing $\frac{1}{1 + x^2}$ in an integral should make you think "arctangent". That forward connection is one reason the AP exam keeps these on the no-calculator section: fluency here pays off again in integration.

A note on domains and the secant case

The square-root forms require $|x| < 1$ for $\arcsin$ and $\arccos$ (their domain), and the secant-family derivatives carry an absolute value $|x|$ because $\text{arcsec}$ is defined for $|x| \geq 1$ and its slope is always positive on each branch. AP Calculus AB rarely tests the secant inverses heavily, so prioritize $\arcsin$, $\arccos$ and $\arctan$, but be aware the $|x|$ exists if a secant inverse appears. When evaluating at a specific point, always check the argument lies in the function's domain; an $\arcsin$ evaluated where $|x| > 1$ has no real value and signals an error upstream.