Differential Calculus

Derivatives and Their Applications

What Is Differential Calculus?

Differential calculus is the branch of mathematics that studies rates of change of functions. It is a fundamental concept used throughout science and engineering, from instantaneous velocity to optimization problems.

Basic concept of derivatives: slope of the tangent line to y=f(x) at x=a equals f'(a) Illustration of a curve y=f(x) with a tangent line at point (a, f(a)). The slope of the tangent line is the derivative f'(a). Basic concept of derivatives: slope of the tangent line x y tangent point slope of tangent = f'(a) y = f(x) The derivative f'(a) is the slope of the tangent line at x = a

This series covers differential calculus systematically in four stages, from high-school introductory level to graduate-level advanced topics.

Content by Level

Introductory

Foundations and applications of derivatives

  • Limits and the definition of the derivative
  • Basic differentiation formulas
  • Product, quotient, and chain rules
  • Trigonometric, exponential, and logarithmic functions
  • Taylor expansion
  • Function analysis and optimization
High school level

Basic

Extension to multivariable calculus

  • Partial and total derivatives
  • Multivariable chain rule
  • Multivariable extrema
  • Lagrange multipliers
  • Introduction to differential equations
  • Rigorous treatment with epsilon-delta
Undergraduate years 1-2

Intermediate

Vector analysis and differential equations

  • Gradient, divergence, and curl
  • Differential geometry of curves and surfaces
  • Second-order linear differential equations
  • Series solutions
  • Systems of differential equations
  • Fundamentals of dynamical systems
Undergraduate years 3-4

Advanced

Deepening and generalizing analysis

  • Differential forms
  • Partial differential equations
  • Sobolev spaces and weak solutions
  • Calculus of variations
  • Introduction to Riemannian geometry
  • Connections to modern analysis
Graduate level
Concept map of differential calculus: four stages and relationships among key topics Concept map of differential calculus, showing the four learning levels (Introductory, Basic, Intermediate, Advanced) and the relationships among limits, continuity, the mean value theorem, Taylor expansion, partial derivatives, and differential equations. Intro Limits & derivatives Basic Formulas Intermediate Theorems & applications Advanced Multivariable & rigor Derivative f'(a) Limits Continuity Mean value thm. Taylor expansion Partial derivatives Diff. equations

Basic Formulas Covered in the Introductory Level

Basic Definition

$$f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$

Power Function

$$(x^n)' = nx^{n-1}$$

Exponential & Logarithmic

$$(e^x)' = e^x$$

$$(\ln x)' = \dfrac{1}{x}$$

Trigonometric Functions

$$(\sin x)' = \cos x$$

$$(\cos x)' = -\sin x$$

Product Rule

$$(fg)' = f'g + fg'$$

Chain Rule

$$(f \circ g)' = (f' \circ g) \cdot g'$$

Reference

Vector Analysis Formula Sheet

Differential operators and integral theorems for 3D vector fields

  • Gradient, divergence, and curl
  • Vector identities (50+ formulas)
  • Formulas in cylindrical and spherical coordinates
  • Divergence theorem and Stokes' theorem
Intermediate to Advanced

Vector and Matrix Calculus

Differentiation of functions with vector and matrix variables

  • Gradient, Jacobian, and Hessian matrices
  • Derivatives of trace, determinant, and inverse
  • Denominator and numerator layout conventions
  • Automatic differentiation and applications
Advanced

Interactive Demos

Gradient Flow Visualization

Visualize the gradient field of a two-variable function in real time. Observe how particles flow in the direction of the negative gradient and converge to local minima.

Basic to Intermediate

Columns

History of Differential Calculus

From ancient Greece through Newton and Leibniz, to the rigorous epsilon-delta formulation and modern automatic differentiation -- tracing 2,000 years of "how to capture change."

Prerequisites

  • Introductory: Basic middle-school mathematics, concept of functions
  • Basic: Introductory content, basics of linear algebra
  • Intermediate: Basic content, linear algebra (matrices, eigenvalues)
  • Advanced: Intermediate content, real analysis, linear algebra

Frequently Asked Questions

What topics does the Derivatives section cover?

The Derivatives section is organized into Intro (conceptual overview), Basic (multivariable calculus, ODE introduction), Intermediate (vector analysis, curves and surfaces, ODEs), and Advanced (differential forms, PDEs, Sobolev spaces, calculus of variations). Specialized subsections on matrix calculus and vector analysis are also available.

What is the relationship between differentiation and integration?

The Fundamental Theorem of Calculus states $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$, making differentiation and integration inverse operations. In higher dimensions, this generalizes to Stokes' theorem $\int_M d\omega = \int_{\partial M}\omega$, unifying gradient, curl, and divergence theorems in a single framework using exterior calculus.

What is matrix calculus and why is it important?

Matrix calculus systematizes differentiation of scalar, vector, and matrix-valued functions with respect to vector or matrix arguments. Rules such as $\frac{\partial}{\partial \mathbf{x}}(\mathbf{x}^T A\mathbf{x}) = (A+A^T)\mathbf{x}$ underpin backpropagation in deep learning, Kalman filtering, optimal control, and numerical optimization.