Learn Why the Math Works
Calculators give answers; these pages explain the reasoning behind them. Each article walks through a key result — definition, proof, worked example — so you understand not just how to apply a rule, but why it is true.
Proof and Explanation Pages
Derivative Rules from First Principles Calculus
f′(x) = lim_{h→0} [f(x+h) − f(x)] / h The limit definition, power rule, sum rule, and constant multiple rule — all derived from scratch. Proof of the Product Rule Calculus
(fg)′ = f′g + fg′ Why the derivative of a product contains two terms and not just one. Proof of the Chain Rule Calculus
(f∘g)′(x) = f′(g(x)) · g′(x) How to differentiate composite functions and the most common mistakes to avoid. Proof of the Quotient Rule Calculus
(f/g)′ = (f′g − fg′) / g² Derived from the product rule via f = g·h, with the condition g(x) ≠ 0. Why eˣ Differentiates to Itself Calculus
d/dx eˣ = eˣ The limit definition of e, the Taylor series for eˣ, and why this property is unique. Matrices as Linear Transformations Linear Algebra
A·v = v₁·col₁(A) + v₂·col₂(A) Columns as images of basis vectors, matrix-vector multiplication, and geometric intuition. Vector Spaces, Span, and Basis Linear Algebra
span{u, v} = {au + bv | a, b ∈ ℝ} Axioms of vector spaces, linear combinations, linear independence, and what a basis is. Eigenvalues and Eigenvectors: Intuition Linear Algebra
Av = λv ↔ det(A − λI) = 0 Directions preserved by a transformation, scaling factors, and a 2×2 worked example. How to Use These Pages
- Read the definition — each page states the result precisely before proving it.
- Follow the proof — every algebraic step is shown and the key idea is highlighted.
- Try the worked example — apply the result to a concrete function before moving on.
- Use the linked calculator — verify your own computations numerically.
Adapted and rewritten from Silas Maths source notes; formulas reviewed for CalxSolver publication.