Code Scholars Tutoring · AP Calculus BC

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

6 topics across 3 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.

Topics 3.1, 3.5

Compositions and choosing derivative rules

Identify outer and inner functions before differentiating. Differentiate the outer function at the inner expression, then multiply by the inner derivative. Several layers require several factors. Combine the chain rule with product or quotient rules when the top-level structure requires them. Exponential bases other than $\displaystyle e$ contribute a logarithmic factor. For a logarithm with base $\displaystyle a>0$ , $\displaystyle a\ne1$ , differentiate $\displaystyle \log_a x$ as $\displaystyle 1/(x\ln a)$ on $\displaystyle x>0$ .

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Topics 3.2, 3.6

Implicit differentiation and tangent equations

When a relation defines $\displaystyle y$ implicitly as a function of $\displaystyle x$ , differentiating a term involving $\displaystyle y$ introduces $\displaystyle y'$ . Solve algebraically for the first derivative. To obtain a second derivative, differentiate the first-derivative relation with product, quotient, and chain rules, then substitute known coordinates and slope. A point must satisfy the original relation.

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Topics 3.3, 3.4

Inverse functions and inverse trigonometry

An inverse derivative uses the slope of the original function at the corresponding input, not at the same numeric argument. Require local invertibility and a nonzero original derivative. The derivatives of inverse trigonometric functions can be derived implicitly; apply the chain rule to their arguments. The principal branches determine the relevant domains.

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