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AP Calculus AB/BC · Advanced Placement (AP) · Unit 2 of 10 · Gatekeeper unit

2Unit 2: Differentiation: Definition & Fundamental Properties

Work through these 10 topics in order. Clear the gatekeeper and the rest of the course opens up.

Contents

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  1. 2.1
    2.1 Average and instantaneous rate at a point

    The average rate of change over [a,a+h][a,a+h] is the slope of the SECANT line through the two points.

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  2. 2.2
    2.2 Defining the derivative (first principles) & notation

    The DERIVATIVE is exactly that limiting instantaneous rate, packaged as one object.

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  3. 2.3
    2.3 Estimating the derivative at a point

    With only a graph or a table (no formula), estimate the tangent slope from nearby points.

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  4. 2.4
    2.4 Differentiability and continuity

    A function that has a tangent slope everywhere cannot jump or break — so differentiable forces continuous.

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  5. 2.5
    2.5 The power rule

    For a power of xx, differentiating just brings the exponent down front and drops it by one — a shortcut proved from the definition.

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  6. 2.6
    2.6 Constant, sum, difference & constant-multiple rules

    Differentiation is linear: it passes through sums and pulls out constant factors, and a constant has zero slope.

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  7. 2.7
    2.7 Derivatives of sin, cos, eˣ, ln x

    A few fundamental derivatives to memorize — the slope of sin⁡\sin is cos⁡\cos, and exe^x is its own slope.

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  8. 2.8
    2.8 The product rule

    A product changes because EACH factor changes — hold one fixed while the other varies, then swap.

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  9. 2.9
    2.9 The quotient rule

    Differentiate a fraction: top-rate times bottom minus top times bottom-rate, all over the bottom squared — order and sign matter.

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  10. 2.10
    2.10 Derivatives of tan, cot, sec, csc

    Build the remaining trig derivatives from sin⁡\sin and cos⁡\cos using the quotient rule.

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