Related Rates, Linear Approximations, and Differentials

When a bond's yield changes, its price changes, and the duration of every bond in the portfolio changes simultaneously. These quantities are not independent: they are related by the pricing formula, so their rates of change with respect to time are related by the chain rule. This is the structure of a related rates problem — multiple quantities depend on a common parameter (usually time), linked by a geometric or algebraic constraint. Taking the derivative of the constraint with respect to the parameter and solving for the unknown rate gives the answer. The second part of the note introduces linear approximations and differentials: the idea that near a point, a function is well approximated by its tangent line, and that a small change $dx$ in $x$ causes a change $dy \approx f'(x)\,dx$ in $y$. This is the foundation of delta-hedging, error propagation, and sensitivity analysis.

Learning objectives. By the end of this note you should be able to: