Selecting procedures for calculating derivatives — AP Calculus BC
1. What Is Selecting Derivative Procedures? ⏱ 3 min
This is a core skill in Unit 3 of the AP Calculus BC CED, worth approximately 9–13% of the total AP exam score, appearing in both multiple-choice and free-response sections. The skill is not just memorizing derivative rules—it is recognizing the form of the function or relation you are working with, then choosing the correct technique to differentiate efficiently and accurately.
2. Selecting the Chain Rule for Composite Functions ★★☆☆☆ ⏱ 4 min
The chain rule is always the correct procedure for differentiating composite functions. For multiple nested composites, apply the chain rule repeatedly, one layer at a time. In Leibniz notation, the rule makes the intuition clear: the total rate of change of $y$ with respect to $x$ is the product of the rate of change of $y$ with respect to the inner function $u$, and the rate of change of $u$ with respect to $x$.
\frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x)
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
Exam tip: Always check for a nested inner function before applying a basic derivative rule—if the base/argument of your function is anything more complicated than just $x$, you will almost always need the chain rule.
3. Selecting Implicit Differentiation for Implicit Relations ★★★☆☆ ⏱ 4 min
Implicit differentiation is the correct procedure for these cases. The method relies on the chain rule: since $y$ is a function of $x$, any term containing $y$ is a composite function of $x$, so you multiply by $\frac{dy}{dx}$ when you differentiate that term. The general steps are: differentiate both sides of the equation with respect to $x$, collect all terms with $\frac{dy}{dx}$ on one side, factor out $\frac{dy}{dx}$, then solve for $\frac{dy}{dx}$.
Exam tip: If a question asks for the derivative at a specific point, substitute the point values into your derivative expression immediately after solving for $\frac{dy}{dx}$—you do not need to simplify further, which saves time.
4. Selecting the Inverse Function Derivative Rule ★★★☆☆ ⏱ 3 min
If $g(x) = f^{-1}(x)$ is the inverse function of $f(x)$, the inverse function derivative rule lets you find the derivative of $g(x)$ without having to solve for the inverse explicitly. The rule comes from implicit differentiation, and it also gives us the standard derivatives for inverse trigonometric functions. When you have an inverse trigonometric function of a non-$x$ argument, you combine the inverse derivative rule with the chain rule.
\frac{d}{dx}\left[f^{-1}(x)\right] = \frac{1}{f'\left(f^{-1}(x)\right)}
This rule is only valid when $f'(f^{-1}(x)) \neq 0$.
Exam tip: You never need to solve for the inverse explicitly to find its derivative at a point—AP problems are designed so the required $x$-value for the original function will always be a simple integer.
5. AP Style Concept Check ⏱ 4 min
Common Pitfalls
Why: Students only differentiate the outer function and stop, confusing composite functions with functions of $x$ directly.
Why: Students treat $y$ as a constant or independent variable, instead of a function of $x$.
Why: Students mix up the order when memorizing the formula.
Why: Students default to implicit differentiation when it is not needed after learning the technique.
Why: Students memorize the inverse trig derivative for $x$ and stop.
Why: Students do not remember the inverse derivative rule works without an explicit inverse.