The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain-rule differentiation.
Environment-dependent bond orders associated with atomic pairs and their derivatives are reused extensively with the aid of linked-list cells to minimize the computation associated with atomic n-tuple interactions (n⩽4 explicitly and ⩽6 due to chain-rule differentiation).
The tool uses a combination of the Adjoint method, the chain rule for differentiation, and the automatic differentiation to compute the sensitivities.
Using Leibniz's generalization of the product rule for differentiation to differentiate both sides of Z ( s ) in (2.8) with respect to s, n times, and taking the limit s → 1, Z ( s ) being analytic at s = 1 on the resulting identity, and finally using the α j in (2.9) and the relation (1.20), we obtain an integral formula for γ n asserted by Theorem 2 below.
Using Leibniz's generalization of the product rule for differentiation when we differentiate the last formula k times and taking the limit s → 1 on the resulting identity, and applying (1.22) to the last resulting formula, we obtain α k + 1 = − ∑ j = 0 k − 1 ( k j ) ψ ( k − j ) ( 1 ) + γ α k = ∑ j = 0 k − 1 ( − 1 ) k − j k ! j !
The chain rule for differentiation reads.
By the chain rule of differentiation, we have T ′ = ∂ T ∂ λ + ∂ T ∂ s α ′.
Transformation from physical to computational domain is carried out using linear transformation and chain rule of differentiation.
This student is not yet at a schema level of APOS Theory regarding application of the sum and chain rule in differentiation of trigonometric functions.
Applying Leiniz's generalization of the product rule for differentiation to (2.20), similarly as in Theorems 1 and 2, we get an integral representation for γ n given in Theorem 3 below.
The use of APOS theory as a framework revealed that several students' errors might be caused by over-generalisation of mathematical rules and properties such as the power rule of differentiation and distributive property in manipulation of algebraic expressions.
