---
tags:
  - mathematics
  - calculus
  - derivatives
  - integrals
created: 2026-05-16
breadcrumb: "Mathematics > Calculus > Calculus essentials"
---

# Calculus essentials

> **Mechanism:** Limits, derivatives, integrals

## Items

- **Limit** — limₓ→a f(x) = L — what f approaches as x approaches a.
- **Derivative** — f'(x) = limₕ→0 (f(x+h) − f(x)) / h — instantaneous rate of change.
- **Power rule** — d/dx (xⁿ) = n x^{n−1}.
- **Product / chain rule** — (fg)' = f'g + fg'.  (f(g(x)))' = f'(g(x)) · g'(x).
- **Indefinite integral** — ∫ f(x) dx — antiderivative. d/dx (∫ f dx) = f.
- **Definite integral** — ∫ₐᵇ f(x) dx — signed area under f from a to b.
- **Fundamental theorem** — If F'(x) = f(x), then ∫ₐᵇ f dx = F(b) − F(a).
- **Partial derivative** — ∂f/∂xᵢ — derivative with respect to one variable, others fixed.
- **Gradient** — ∇f — vector of partial derivatives; points in direction of steepest ascent.
- **ODE / PDE** — Equations involving derivatives. Ordinary: one variable. Partial: many.

## Related

- [[linear-algebra]]
- [[famous-math-equations]]