Unit 2: Differentiation: Definition & Fundamental Properties
Work through these 10 topics in order. Clear the gatekeeper and the rest of the course opens up.
2.1 Average and instantaneous rate at a point
The average rate of change over [a,a+h] is the slope of the SECANT line through the two points.
2.2 Defining the derivative (first principles) & notation
The DERIVATIVE is exactly that limiting instantaneous rate, packaged as one object.
2.3 Estimating the derivative at a point
With only a graph or a table (no formula), estimate the tangent slope from nearby points.
2.4 Differentiability and continuity
A function that has a tangent slope everywhere cannot jump or break — so differentiable forces continuous.
2.5 The power rule
For a power of x, differentiating just brings the exponent down front and drops it by one — a shortcut proved from the definition.
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.
2.7 Derivatives of sin, cos, eˣ, ln x
A few fundamental derivatives to memorize — the slope of \sin is \cos, and e^x is its own slope.
2.8 The product rule
A product changes because EACH factor changes — hold one fixed while the other varies, then swap.
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.
2.10 Derivatives of tan, cot, sec, csc
Build the remaining trig derivatives from \sin and \cos using the quotient rule.