Code Scholars Tutoring · AP Calculus BC
10 topics across 4 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
Topics 2.1, 2.2, 2.3
The derivative is the limit of secant slopes when that limit exists. It is both the slope of the tangent and an instantaneous rate with output-units per input-unit. Estimate from nearby table entries; a symmetric secant is often useful but still approximate. Distinguish a derivative value at one point from a derivative function on a domain.
Topics 2.4
Differentiability at a point implies continuity there; the converse fails. Corners have different one-sided slopes, cusps may have unbounded one-sided slopes, and vertical tangents have no finite derivative. Discontinuities automatically prevent differentiability. For a piecewise function first enforce continuity, then compare the limiting derivative from each side.
Topics 2.5, 2.6, 2.7, 2.10
Use the power rule on powers, including negative and fractional powers where defined. Differentiate constants to zero and distribute differentiation through finite sums. Memorize derivatives of the six trigonometric functions and the natural exponential and logarithm. Trigonometric formulas assume radians; domain restrictions remain in force after differentiating.
Topics 2.8, 2.9
A product derivative differentiates one factor at a time and adds the two terms. For a quotient, multiply the denominator by the numerator derivative, subtract numerator times denominator derivative, and divide by the denominator squared. Simplification before differentiating may shorten the work. A derivative formula is valid only where the original function and relevant derivatives exist.
