Overview

Differentiation is the mathematical technology behind every Greek letter on a trading desk. Delta, gamma, vega, rho, and theta are all derivatives — they tell a risk manager how much a position's value changes when the underlying variable moves. This topic builds differentiation from the ground up: from the limit definition (which also defines the instantaneous rate of change of any quantity), through the mechanical rules that make differentiation tractable, to the special applications in finance and science where the derivative is the central object.

The topic is structured in two arcs. The first arc (Lessons 1–5) builds the toolkit: the limit definition and derivative function, the basic differentiation rules (polynomials, exponentials, products, quotients, trigonometric functions), the chain rule, and implicit and logarithmic differentiation. The second arc (Lessons 6–8) applies the toolkit: rates of change in finance, exponential growth and decay, and the geometric tools of related rates, linear approximation, and differentials.

• compute derivatives from the limit definition and recognise the derivative as a rate of change;
• apply the power, product, quotient, and chain rules to differentiate any expression built from polynomials, exponentials, logarithms, and trigonometric functions;
• differentiate implicitly and use logarithmic differentiation for products of many factors or for $x^x$-type expressions;
• compute marginal cost/revenue/profit and interpret them as instantaneous rates of change;
• model exponential growth and decay (radioactive decay, continuous compounding) using the ODE $y' = ky$;
• find related rates by implicit differentiation of a geometric or physical constraint with respect to time;
• write a linear approximation and understand differentials as first-order error estimates.

1. Contents
2. The big picture