Skip to main content (Press Enter)
Pages similar to:
Partial derivative examples
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 by limit definition
Description with example of how to calculate the partial derivative from its limit definition.
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.
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.
Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables.
The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable.
Non-differentiable functions must have discontinuous partial derivatives
A visual tour demonstrating discontinuous partial derivatives of a non-differentiable function, as required by the differentiability theorem.
A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable.
Elementary partial derivative problems
Sample problems illustrating the partial derivative.
Solutions to elementary partial derivative problems
Solutions to sample problems illustrating the partial derivative.
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.
Introduction to the multivariable chain rule
Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.
Multivariable chain rule examples
Examples demonstrating the chain rule for 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.
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.
Partial derivative examples