Faà di Bruno Partition Derivatives

Organize higher chain-rule derivatives by set partitions and Bell multiplicities.

Faà di Bruno Partition Derivatives

The generalized chain rule, known as Faà di Bruno's formula, provides a way to calculate higher-order derivatives of composite functions. Instead of manually taking the derivative layer by layer and relying heavily on the product rule, the formula organizes the terms based on partitions of the set of derivatives. These partitions determine the coefficients, which can be computed systematically using partial Bell polynomials.

Worked example: computing the third derivative

Let g(x) = x² + 1 and f(u) = exp(u). We want to compute the third derivative of the composite function (f o g)(x) at x = 0.5. First, let's find the inner values and derivatives at x = 0.5:

g(0.5) = (0.5)² + 1 = 1.25
gPrime(0.5) = 2(0.5) = 1
gSecond(0.5) = 2
gThird(0.5) = 0

The outer function is the exponential, so its derivatives evaluated at u = 1.25 are all f^(k)(1.25) = exp(1.25) ≈ 3.4903429574618414. We compute the third derivative (f o g)^(3) using the Bell polynomials B_(3,k):

For k=1, B_(3,1) = gThird = 0. Contribution: 0.
For k=2, B_(3,2) = 3 * gPrime * gSecond = 3(1)(2) = 6. Contribution: 6 * exp(1.25) ≈ 20.9420577.
For k=3, B_(3,3) = (gPrime)³ = 1³ = 1. Contribution: 1 * exp(1.25) ≈ 3.4903429.

The sum of these contributions yields the total composite derivative: 0 + 20.9420577 + 3.4903429 = 24.4324007. This precisely organizes the multiple chain rule terms according to how many factors of the outer derivative appear.

Experiment with Bell multiplicities

Open Faà di Bruno Partition Derivatives. You can interactively change the inner and outer functions and observe how individual partition contributions sum to the total derivative.

Set Outer function to "Exponential", Inner offset to 1, and Probe x to 0.5.
Set Derivative order to 3 and Partition count to 2.
Check that the Selected Bell polynomial is 6 and the Selected contribution is approximately 20.9420577, verifying the calculation above.
Change Derivative order to 4 and leave Partition count at 2. Notice how the polynomial grows to include more partition combinations, specifically B_(4,2) = 4 * gPrime * gThird + 3 * (gSecond)² = 0 + 3(2)² = 12.

Domain restrictions and vanishing derivatives

The logarithm function requires a strictly positive argument. If g(x) ≤ 0, the composite function and its derivatives are undefined, and the UI will report "Unavailable". The underlying Bell polynomials, which depend only on the inner function's derivatives, remain mathematically defined but are separated from the undefined outer derivatives.

Additionally, if the outer function is a low-degree polynomial like the cube function, its derivatives above the third order will identically vanish. Consequently, higher-order partition contributions evaluate to zero, regardless of the complexity of the inner Bell polynomials.

References for the formula

The standard definition and examples of this generalized chain rule using partial Bell polynomials can be found onWolfram MathWorld: Faà di Bruno's Formula.

Faà di Bruno Partition Derivatives