Manipulating differentials - references?

[Ultimâ]

I need to confirm Equations 1.8, 1.10, 1.12, 1.15 and 1.17 are correct (in the pdf), can anyone suggest some suitable references (either books or web links) that describe the manipulations I've used? Some info about the material in the pdf below:

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In the pdf document, equation 1.1 represents the complete form of x(k)=f(x(k-1)) + w(k-1) where it uses fx to represent f(x(k-1)).

Equations 1.5 and 1.7, or their inverse, are used to find most of the derivatives in the Jacobian matrix represented in Equation 1.3.

 

[gtoguy97]

Equation 1.8 is derived from Equation 1.3 by using the chain rule and the fact that x(k) is a function of x(k-1).Equations 1.10, 1.12, 1.15 and 1.17 are also derived from Equation 1.3, by applying the chain rule and the fact that x(k) is a function of x(k-1).----------------------------------------------------------------------------------------One possible reference for the manipulations used in equations 1.8, 1.10, 1.12, 1.15 and 1.17 is the book "Calculus: Early Transcendentals" by James Stewart. This book provides a comprehensive overview of calculus topics such as derivatives, chain rule, and Jacobian matrices. Additionally, it provides examples to illustrate the concepts discussed in the book. Another possible reference is Khan Academy’s website. Khan Academy offers a range of free tutorials and practice questions related to calculus topics, including derivatives, chain rule, and Jacobian matrices. It is an excellent resource for learning and reinforcing these concepts.