Differentiation: Definition and Fundamental Properties — AP Calculus BC
1. Unit at a Glance
This unit follows a logical progression from first principles to practical differentiation. We start by connecting familiar ideas of rate of change to the formal definition of the derivative, then build up the core rules that let you quickly compute derivatives without relying on limits for every calculation.
By the end of the unit, you will be able to differentiate all basic elementary functions, which prepares you for more advanced differentiation techniques and applications of derivatives in later units.
Common Pitfalls
Why: Continuity does not guarantee differentiability. Functions with corners, cusps, or vertical tangents are continuous but not differentiable.
Why: Many students incorrectly simplify $(fg)' = f'g'$ or misorder terms in the quotient rule, leading to wrong results.
Why: The power rule only applies to functions with constant exponents and variable bases, not the reverse.