Overview

Differentiation is a central notion of the calculus. It began simply as a method to find the tangent to a curve in the plane. The notion was then extended to include tangent planes to surfaces and was developed into numerous definitions and types of mathematical objects.

Example

A graph of the function {% y = x^2 %} with the tangent line drawn.


Moving the slider moves the tangency point.

Definition

The definition of differention relies on the notion of a limit. The derivative of a function {% f(x) %} is a new function , here denoted by {% f'(x) %} where if it exists.

Topics

• Analytic Techniques : a set of analytic techniques for computing the derivative
• Abstract Definition (Total Derivative)
  • Gateaux Derivative
  • Frechet Derivative
• Multi-Variable
  • Differential Forms: are used to extend calculus to be used on curved surfaces and manifolds.
  • Directional Derivative
• Functional Differentiation - gives a notion of differentiating a functional by a function.
• Theorems
  • Mean Value Theorem
• Numeric Implementation