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Intermediate Value Theorem, location of roots
Using the Intermediate Value Theorem to find small intervals where a function must have a root.
Multivariable chain rule examples
Examples demonstrating the chain rule for multivariable 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.
Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case.
The gradient vector
The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector.
The idea of the derivative of a function
The derivative of a function as the slope of the tangent line.
Derivatives of polynomials
How to compute the derivative of a polynomial.
Derivatives of more general power functions
How to compute the derivative of power functions.
A refresher on the quotient rule
How to compute the derivative of a quotient.
A refresher on the product rule
How to compute the derivative of a product.
A refresher on the chain rule
How to compute the derivative of a composition of functions.
Calculating one derivative in terms of another derivative.
Derivatives of transcendental functions
A list of formulas for taking derivatives of exponential, logarithm, trigonometric, and inverse trigonometric functions.
A way to simplify evaluation of limits when the limit is an indeterminate form.
The second and higher derivatives
Taking derivatives multiple times to calculate the second or higer derivative.
Inflection points, concavity upward and downward
Finding points where the second derivative changes sign.
Approximating a nonlinear function by a linear function
The secant line and tangent line are two ways to approximate a nonlinear function by a linear one.
Developing intuition about the derivative
An intuitive exploration into the properties of the derivative, illustrated by interactive graphics.