A Counterexample: Function Continuity

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Jacobian Matrix: Derivative of Composite Functions

Explains the Jacobian matrix and derivative of composite functions with examples.

Partial Derivatives: Introduction

Introduces partial derivatives in functions of several variables and their properties.

Theorems of Continuity and Derivative Number

Explores the theorems of continuity and derivative numbers, including strict monotonicity and the Newton ratio.

Derivative of an Integral

Covers the derivative of an integral dependent on a parameter and its fundamental properties.

Differential Functions: Derivability and Gradient

Explores differentiability in functions and the role of gradients in directional derivatives.

Change of Coordinates: Invertible Matrices & Functions

Explores invertible matrices, injective functions, and the theory of reciprocal functions.

Partial Derivatives: Definitions and Applications

Explores the definitions and applications of partial derivatives in functions of several variables, emphasizing the importance of specifying the choice of variables.

Derivability and Chain Rule

Covers the demonstration of the chain rule and the theorem of Rolle.

Partial Derivatives and Derivability Theorems

Introduces partial derivatives, directional derivatives, and differentiability of functions with examples and the Schwartz theorem.

Computing Derivatives: Derivatives and Algebraic Operations

Covers the computation of derivatives and the properties of differentiable functions.