Derivation: finite differences from Taylor
How the truncated Taylor expansion yields the forward, backward and central finite differences of the first derivative.
Derivations
How each formula is obtained, step by step.
Foundations
Interpolation
Derivation: Newton coefficients via divided differences
Where and come from when forcing the polynomial through the points, and how the recursion generates every higher-order coefficient.
Derivation: Lagrange basis functions
How the requirement to equal 1 at one node and 0 at the others forces the product form of the functions.
Differentiation
Derivation: three-point forward O(h²)
How to combine two Taylor expansions to eliminate the second derivative and get an order-2 forward difference for the first derivative.
Derivation: Richardson raises the order
From the error form to the extrapolation formula: why combining and removes the term.
Integration
Derivation: trapezoid rule
Integrating the linear Lagrange interpolant on [a,b] to obtain the trapezoid rule and its geometric interpretation.
Derivation: composite trapezoid
From summing simple trapezoids on n subintervals to the weights 1,2,...,2,1 and the global error.
Derivation: simple midpoint
Replacing the function by a central height to obtain the centred rectangle rule and its error.
Derivation: composite midpoint
Generalising midpoint to n subintervals by using the centre of each block and summing local errors.
Derivation: Simpson 1/3
From three equally spaced nodes to Simpson's 1, 4, 1 weights by integrating the quadratic Lagrange polynomial.
Derivation: two-point Gauss-Legendre
How imposing exactness up to degree 3 gives the nodes and weights 1 on .
ODEs
Derivation: Euler's method and its order
Three independent routes lead to Euler's formula (Taylor, incremental quotient and integration), and the analysis of the Taylor remainder proves the method is first order.
Derivation: implicit Euler
Approximating the derivative at the new node with a backward difference produces the implicit Euler method and the nonlinear equation to solve at each step.
Derivation: Heun's method
Full derivation of Heun: via the Taylor expansion to second order combined with the two-variable Taylor of f, and via the trapezoidal rule with an Euler prediction.
Derivation: fourth-order Runge-Kutta
Applying Simpson's rule to the integral form of the IVP and approximating the unknown slopes with chained internal evaluations produces the classical RK4 and explains its 1, 2, 2, 1 weights.
Derivation: two-step Adams-Bashforth (AB2)
Full construction of AB2: integral form of the IVP, Lagrange interpolant of f at the two previous nodes, change of variable, integrals computed term by term, local error and generalization to AB3 and AB4.
Derivation: one-step Adams-Moulton (AM2)
Full construction of AM2: Lagrange interpolant including the new node, change of variable, computed 1/2-1/2 weights, connection with the trapezoidal rule and local error.
Linear systems
Derivation: Jacobi iteration matrix
From solving each unknown out of its equation to the matrix form x=−D⁻¹(L+U)x+D⁻¹b.
Nonlinear equations
Derivation: Newton-Raphson and its quadratic order
Newton's formula via the tangent line and via the Fundamental Theorem of Calculus, plus the complete Taylor proof that its error equation is quadratic.
Derivation: bisection error bound
Why the bisection error halves at every iteration, and how to predict in advance how many iterations a given tolerance requires.
Derivation: fixed-point convergence and order
Expanding by Taylor around the fixed point yields the method's error equation and proves both the criterion and the order theorem.
Nonlinear systems
Derivation: Newton for systems by linearization
The first-order multivariate Taylor expansion linearizes F around the current iterate; setting that linearization to zero gives the Newton step, with the Jacobian in the role of the derivative.