Math Insight

Pages similar to: Multivariable chain rule examples

• Special cases of the multivariable chain rule
  Illustrations of different special cases of the multivariable chain rule and their relationship to the general case.
• Introduction to the multivariable chain rule
  Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.
• 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.
• 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.
• The gradient vector
  The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector.
• Newton's Method
  A method to approximate the roots to an equation.
• Matrices and determinants for multivariable calculus
  Brief description of matrices and determinants, at the level needed for multivariable calculus.
• 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.

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