Given a multivariable function (a surface in three-dimensional space), the directional derivative in the direction of a vector represents the slope of the function (how fast the function is changing), at a particular point, in the direction that the vector is pointing.

In this video, we'll focus on creating tree diagrams for Case I (one independent variable) and Case II (multiple independent variables) multivariable functions. From the tree diagrams, we'll construct formulas to represent the derivative(s) of each multivariable function.

This video describes how to use chain rule to find the derivative of a multivariable function, where the independent variables from the multivariable function are each defined in terms of one or more parameter values.

Finding the linear approximation equation of a function at a point is the same as finding the equation of the tangent line to the function at the same point.

The tangent line to a single variable function was the line in two-dimensional space that represented the slope of the single variable function. Similarly, the tangent plane to a multivariable function is the plane in three-dimensional space that represents the slope of the multivariable function. To find the equation of the tangent plane, you need The slope of the function at the point of tangency in the direction of each independent variable (the partial derivative of the function with respect to each variable, evaluated at the point of tangency) The point of tangency

This video describes how to find the second-order partial derivatives of a multivariable function.

You can find partial derivatives for a multivariable function, now matter how many variables are involved in the function. If there are three variables in the function, you'll have three partial derivatives, one with respect to each of the variables. If there are five variables in the function, you'll have five partial derivatives, one with respect to each of the variables.

Taking the derivative of a single variable function is easy, because you simply have to differentiate with respect to the only variable in the function, and therefore, there's only one derivative. Multivariable functions are slightly more complicated, because you have to find a different derivative for each of the variables in the function. These are called partial derivatives, and you'll have one partial derivative for each of the variables in the function. For example, given a function z defined in terms of x and y, you'll have two partial derivatives, one for each of the variables x and y. You'll call them the partial derivative of z with respect to x and the partial derivative of z with respect to y.

The domain of a multivariable function will be defined in terms of three-dimensional space (3D, or R^3), whereas the domain of a single variable function could be defined in terms of two-dimensional space (2D, or R).

The Maclaurin series of a function is the same as the function's Taylor series. We just give the Maclaurin series a special name because it's always centered about a=0, whereas a Taylor series can be centered about any value.