Intuitive Approach to Derivatives

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Differential Functions: Definitions and Notations

Covers the definition of differentiable functions and introduces the concept of differential functions.

Connections: motivation and definition

Explores the definition of connections for smooth vector fields on manifolds.

Advanced Analysis II: Differentiability and Continuity

Explores differentiability and continuity in advanced analysis with examples and counterexamples.

Definition of Differentiable Functions

Covers the definition of differentiable functions and explores the representation of the remainder in derivatives.

Jacobian Matrix: Derivative of Composite Functions

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

Integration by Substitution

Explores integration by substitution with proofs and examples on anti-derivatives and function equivalence.

Differentiation and Coordinate Changes in Multivariable Functions

Discusses differentiation of multivariable functions and coordinate transformations, including polar and cylindrical coordinates, along with the Laplacian operator and its applications.

Derivative of a Function: Tangent Equation

Explores finding tangent equations and slopes through derivatives.

Derivatives and Continuity

Covers derivatives, continuity, properties, and applications of derivative rules.

Partial Derivatives: Understanding Tangents

Explores partial derivatives and their role in determining tangents to graphs.