Differentiation: Composite, Implicit, and Inverse Functions — AP Calculus BC
1. Unit at a Glance
This unit follows a logical progression from extending basic differentiation to handling increasingly complex function types. We start with the chain rule, the core tool for differentiating composite functions, then move to implicit differentiation for relations that cannot be solved explicitly for the dependent variable. We then apply these techniques to inverse functions (including inverse trigonometric functions), learn to calculate higher-order derivatives, and wrap up with practice selecting the right procedure for any differentiation problem.
Common Pitfalls
Why: This is the most frequent mistake on differentiation problems, leading to missing terms and incorrect coefficients.
Why: Mixing up the input and output of the inverse function leads to incorrect final results.
Why: Higher-order derivatives require repeated differentiation, so the chain rule applies at every step.