Derivatives — Basic

Extending Differentiation to Several Variables (University Year 1–2)

Overview of the Basic Level

The basic level covers the extension from single-variable functions to functions of several variables. From the concepts of partial and total differentials, it treats extrema of multivariable functions, constrained optimization, and an introduction to differential equations.

Learning objectives

  • Understand the concepts of partial and total differentials
  • Use the chain rule for multivariable functions
  • Classify extrema of multivariable functions with the Hessian matrix
  • Master the method of Lagrange multipliers
  • Understand the multivariable Taylor expansion
  • Solve basic differential equations

Contents

  1. Chapter 1 Partial Derivatives

    Definition of partial derivatives, computing partial derivatives, higher-order partial derivatives

  2. Chapter 2 Total Differential and Tangent Planes

    Definition of the total differential, the equation of the tangent plane, linear approximation

  3. Chapter 3 The Multivariable Chain Rule

    The chain rule formula, the implicit function theorem, change of variables

  4. Chapter 4 Extrema of Multivariable Functions

    Critical points and extrema, the Hessian test, saddle points

  5. Chapter 5 Lagrange Multipliers

    The principle of Lagrange multipliers, optimization with equality constraints, worked examples

  6. Chapter 6 Multivariable Taylor Expansion

    Second-order approximation, the general Taylor expansion, error estimates

  7. Chapter 7 Introduction to Differential Equations

    Separable equations, first-order linear ODEs, worked examples

  8. Chapter 8 Toward Rigorous Analysis

    Rigorous definitions of limits, continuity, and differentiability (the ε–δ method)

  • Differentiation of Parametric Curves — the formula dy/dx=(dy/dt)/(dx/dt), its derivation, second derivatives, and examples such as the cycloid…
  • Saddle Points — the definition of the saddle points that arise in multivariable extrema, the Hessian-sign test, and examples (a supplement to Chapter 4)
  • Vector Fields — a map assigning a vector to each point of space: gradient, divergence, curl, streamlines, and the relationship between conservative fields and potentials

Advanced Single-Variable Differentiation (University Level)

A rigorous theory of single-variable differentiation, one step beyond the introduction (high-school level). Good to read as a follow-up after finishing the introductory course.

  • The Definition of the Derivative — the rigorous limit definition of the derivative, plus the differentiation formulas for basic functions (powers, exponentials, logarithms, trigonometric functions) (a reference)
  • The Mean Value Theorem — rigorous proofs of Rolle's theorem, the mean value theorem, and Cauchy's mean value theorem
  • L'Hôpital's Rule — a rigorous treatment of indeterminate-form limits using Cauchy's mean value theorem
  • Taylor Expansion — Taylor's theorem, the Lagrange remainder, and the radius of convergence
  • Taylor Expansions of Basic Functions — the Maclaurin expansions of $e^x,\sin x,\cos x,\ln(1+x)$ and Euler's formula
  • The Product Rule — a rigorous proof from the definition of the limit and Leibniz's general formula
  • The Quotient Rule — deriving the quotient rule for $f/g$ from the definition of the limit, confirmed with examples
  • Cauchy's Mean Value Theorem — a proof via an auxiliary function and Rolle's theorem, and its application to L'Hôpital's rule

Prerequisites

FAQ

What is covered in Derivatives — Basic?

It covers partial derivatives, total differentials, the multivariable chain rule, extrema of multivariable functions, Lagrange multipliers, multivariable Taylor expansion, an introduction to ordinary differential equations, and rigorous analysis (epsilon–delta). Building on the basics of single-variable differentiation, it broadens understanding toward several variables, applications, and rigor.

What kind of method is the method of Lagrange multipliers?

It is a method for maximizing or minimizing $f(x,y)$ subject to a constraint $g(x,y)=0$. Introducing an auxiliary variable (the multiplier) $\lambda$, one solves $\nabla f = \lambda \nabla g$ together with $g=0$. It is the foundation of optimization with equality constraints and is widely applied in physics, economics, and machine learning.

How is the multivariable Taylor expansion used?

Near a point $(a,b)$, $f(x,y) \approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac{1}{2}[f_{xx}(x-a)^2+2f_{xy}(x-a)(y-b)+f_{yy}(y-b)^2]+\cdots$. It is used for classifying extrema (positive-definiteness of the Hessian) and for local linearization and second-order approximation in numerical computation.

Essays

  • The Day One Slope Is No Longer Enough [Essay] — From a world where "the slope is a single number" to one where "the slope changes with direction." The idea of partial derivatives and the gradient, told as if strolling over a hill.
  • The Magic of Unnesting [Essay] — Unwinding functions nested inside functions by multiplying rates of change like meshing gears. A relaxed look at the idea of the chain rule.
  • Flat Doesn't Mean the Top [Essay] — A zero slope does not guarantee a summit. Is it a valley floor, or a pass (a saddle point)? The idea of how the second derivative distinguishes the "curvature" that the first derivative cannot see, told in the mood of a mountain walk.