Differentiating inverse trigonometric functions — AP Calculus AB
1. Derivation of Basic Inverse Trigonometric Derivative Rules ★★☆☆☆ ⏱ 4 min
All derivative rules for inverse trigonometric functions can be derived using implicit differentiation, leveraging the inverse function definition: if $y = f^{-1}(x)$, then $f(y) = x$ for the restricted range of the inverse function.
For $y = \arcsin x$, we start with $\sin y = x$, where $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, so $\cos y \geq 0$. Differentiating implicitly with respect to $x$ gives:
\cos y \cdot \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{\cos y}
Using the Pythagorean identity $\cos^2 y = 1 - \sin^2 y = 1 - x^2$, we take the positive root to get the final rule:
\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1
Following the same process gives the rules for arccosine and arctangent:
\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1 \\ \frac{d}{dx}\arctan x = \frac{1}{1 + x^2}, \quad \text{all real } x
2. Differentiating Composite Inverse Trigonometric Functions ★★★☆☆ ⏱ 4 min
Nearly all AP exam problems involve composite inverse trigonometric functions of the form $y = \arcsin(u(x))$, $y = \arccos(u(x))$, or $y = \arctan(u(x))$, where $u(x)$ is a non-constant differentiable function. To differentiate these, apply the chain rule: $\frac{dy}{dx} = f'(u(x)) \cdot u'(x)$, where $f$ is the outer inverse trig function.
The general chain rule forms for each core function are:
\frac{d}{dx}\arcsin(u(x)) = \frac{u'(x)}{\sqrt{1 - [u(x)]^2}}, \quad |u(x)| < 1 \\ \frac{d}{dx}\arccos(u(x)) = \frac{-u'(x)}{\sqrt{1 - [u(x)]^2}}, \quad |u(x)| < 1 \\ \frac{d}{dx}\arctan(u(x)) = \frac{u'(x)}{1 + [u(x)]^2}, \quad \text{all real } x
3. Tangent Line Problems for Inverse Trigonometric Functions ★★★☆☆ ⏱ 3 min
A common AP exam application of inverse trig derivatives is finding the equation of a tangent line to an inverse trigonometric function at a given point. This requires three key steps: (1) calculating the correct y-coordinate using the inverse's restricted range, (2) finding the derivative at the point for slope, (3) writing the tangent line in point-slope form, which is preferred unless another form is requested.
4. Mixed AP-Style Worked Examples ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Students memorize the basic rule for $\arcsin x$ and forget composite functions require multiplying by the derivative of the inner function.
Why: Students forget the range restriction on inverse sine that guarantees $\cos y$ is non-negative, so they leave an ambiguous sign.
Why: Students forget the derivative only exists where $|u(x)| < 1$, not $|u(x)| \leq 1$, because the derivative is undefined at the endpoints of the domain.
Why: Students confuse the denominator of the arctangent derivative with the denominator of arcsine, swapping the plus sign for a minus.
Why: Students remember the general inverse function derivative rule but forget to simplify to the standard form in x required on the exam.