Chapter 6
Numerical Differentiation
This chapter develops methods for computing derivatives of functions given analytically or as tabulated data, motivated by the central role of derivatives in physical laws. It begins with differentiation via interpolation and then derives finite-difference formulas from Taylor series, including centered formulas for interior points and one-sided formulas at boundaries. It analyzes how truncation and roundoff errors compete to determine a useful step size and explains why differentiation amplifies noise. Richardson extrapolation is introduced as a systematic way to improve accuracy. The chapter closes by extending finite differences to partial and mixed derivatives and the five-point Laplacian.
In this chapter
- 6.1 From physical laws to numerical differentiation
- 6.2 Differentiation via interpolation
- 6.3 Finite-difference methods from Taylor series
- 6.4 Error analysis and Richardson extrapolation
- 6.5 Extension to multiple variables
- 6.6 Summary and practical guidelines
- 6.7 Computational exercises and projects
Companion resources
Codes
MATLAB, Python, C++, and Fortran programs for differentiation of tabulated data and for Richardson extrapolation.
View Chapter 6 codes →Exercises & Projects
Additional exercises for practice and homework, together with physics projects that use numerical differentiation in larger computational problems.
View Chapter 6 exercises & projects →Resources
Finite-difference formulas, step-size and error guidance, benchmark functions, and selected references.
View Chapter 6 resources →Updates & Errata
Corrections, clarifications, and post-publication updates associated with Chapter 6.
View Chapter 6 updates & errata →