E.1 Differentiation Foundations

The foundations of calculus. The DERIVATIVE f'(x) is a NEW function whose value at any x equals the slope of the tangent to y = f(x) at that x. Geometrically: tangent slope. Physically: instantaneous rate of change (velocity, growth rate, marginal cost). Defined formally as the LIMIT of the difference quotient: f'(x) = lim_{h→0} [f(x+h) − f(x)] / h — set up the quotient, simplify so h CANCELS, then take h → 0. Computing every derivative this way is slow, so the POWER RULE provides a shortcut for any polynomial term: d/dx(axⁿ) = an·x^(n−1) — bring the exponent down, subtract 1. Constants disappear (slope of horizontal line = 0); negative exponents work the same way (sign flips). With f'(x) in hand, READ off behaviour: f'(x) > 0 → f increasing; f'(x) < 0 → f decreasing; f'(x) = 0 → stationary point (max, min, or inflexion). And USE f'(x₀) as a slope to write the TANGENT line y − y₀ = f'(x₀)(x − x₀) and the NORMAL line (perpendicular: m_n = −1/f'(x₀)) at any point on the curve.