Skip to main content

Section 1: Differential Calculus

Overview​

Focus on derivatives, gradients, and higher-order analysis in single and multiple variables, with emphasis on the tools most used in machine learning.

Topics Covered​

Chapter 1: Derivatives and Gradients​

  • Partial derivatives and the gradient vector
  • Directional derivatives and steepest ascent
  • Level curves and gradient perpendicularity

Chapter 2: The Chain Rule​

  • Single-variable chain rule review
  • Multivariate chain rule and total derivative
  • Composition of vector-valued functions

Chapter 3: Jacobians and Hessians​

  • The Jacobian matrix of vector-valued functions
  • The Hessian matrix and second-order conditions
  • Change of variables formula

Chapter 4: Taylor Approximation​

  • Taylor series in one dimension
  • Multivariate Taylor expansion
  • Linearization and quadratic approximation

Learning Objectives​

  • Compute partial derivatives and gradients for multivariable functions
  • Apply the chain rule in multivariate settings
  • Construct and interpret Jacobian and Hessian matrices
  • Use Taylor expansion for local approximation of functions