Skip to content
Calculus I / Differentiation

The Chain Rule

The Black-Scholes delta of a call option involves e−d12/2e^{-d_1^2/2} — the exponential of a squared expression. The Sharpe ratio of a log-return involves ln⁡(σ/rˉ)\ln(\sigma/\bar{r}). Duration of a bond portfolio involves d/drd/dr of a sum of exponentials in rr. All of these are compositions of functions, and none can be differentiated by the rules developed so far. The chain rule is the rule for differentiating compositions: it states that ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x) — differentiate the outer function evaluated at the inner, times the derivative of the inner. After the product rule, the chain rule is the most-used differentiation tool in mathematical finance.
Learning objectives. By the end of this note you should be able to:
  • state the chain rule in both prime notation F′(x)=f′(g(x))⋅g′(x)F'(x) = f'(g(x))\cdot g'(x) and Leibniz form dy/dx=(dy/du)(du/dx)dy/dx = (dy/du)(du/dx);
  • identify the outer and inner functions of a composition and apply the chain rule;
  • apply the "generalised power rule" to (g(x))n(g(x))^n for any real nn;
  • differentiate eg(x)e^{g(x)}, ln⁡g(x)\ln g(x), sin⁡g(x)\sin g(x), and ag(x)a^{g(x)};
  • apply the chain rule repeatedly for iterated compositions.

In this lesson

  1. The chain rule
  2. The generalised power rule
  3. Chain rule for exponentials and logarithms
  4. Trig derivatives with the chain rule
  5. Iterated composition (three or more layers)

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.