Pages similar to: Introduction to the multivariable chain rule
- Multivariable chain rule examples
Examples demonstrating the chain rule for multivariable functions. - Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case. - A refresher on the chain rule
How to compute the derivative of a composition of functions. - The idea of the chain rule
An illustration of the basic concept of the chain rule using interactive graphics to diagram the relevant points on the graphs and the corresponding slopes. - Simple examples of using the chain rule
Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. - Introduction to partial derivatives
The concept of partial derivatives is introduced with an illustration of heating costs. Interactive graphics demonstrate the properties of partial derivatives. - Partial derivative examples
Examples of how to calculate partial derivatives. - Partial derivative by limit definition
Description with example of how to calculate the partial derivative from its limit definition. - Introduction to differentiability in higher dimensions
An introduction to the basic concept of the differentiability of a function of multiple variables. Discussion centers around the existence of a tangent plane to a function of two variables. - The multivariable linear approximation
Introduction to the linear approximation in multivariable calculus and why it might be useful. - Examples of calculating the derivative
Examples showing how to calculate the derivative and linear approximation of multivariable functions. - The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable. - Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics. - The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions. - Derivatives of parameterized curves
The derivative of the vector-valued function parameterizing a curve is shown to be a vector tangent to the curve. - An introduction to the directional derivative and the gradient
Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables. - Directional derivative and gradient examples
Examples of calculating the directional derivative and the gradient. - Derivation of the directional derivative and the gradient
Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions. - Parametrized curve and derivative as location and velocity
Description of a parametrization of a curve as the position of a particle and the derivative as the particle's velocity. Illustrated with animated graphics. - Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables.