Faà di Bruno Partition Derivatives
Organize higher chain-rule derivatives by set partitions and Bell multiplicities.
Free theory articleOrganize higher chain-rule derivatives by set partitions and Bell multiplicities.
Concept & interpretation
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.
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.