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# 14: Partial Differentiation

• • Contributed by David Guichard
• Professor (Mathematics) at Whitman College
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• 14.1: Functions of Several Variables
A function f:R2→R maps a pair of values (x,y) to a single real number. The three-dimensional coordinate system we have already used is a convenient way to visualize such functions: above each point (x,y) in the x - y plane we graph the point (x,y,z) , where of course z=f(x,y).
• 14.2: Limits and Continuity
To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which we needed to understand continuous functions and to define the derivative. Limits involving functions of two variables can be considerably more difficult to deal with; fortunately, most of the functions we encounter are fairly easy to understand.
• 14.3: Partial Differentiation
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
• 14.4: The Chain Rule
Not surprisingly, the same chain rule that was formulated for a function on one variable also works for functions of more than two variables.
• 14.5: Directional Derivatives
The directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
• 14.6: Higher order Derivatives
In single variable calculus we saw that the second derivative is often useful: in appropriate circumstances it measures acceleration; it can be used to identify maximum and minimum points; it tells us something about how sharply curved a graph is. Not surprisingly, second derivatives are also useful in the multi-variable case, but again not surprisingly, things are a bit more complicated.
• 14.7: Maxima and minima
The maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function.
• 14.8: Lagrange Multipliers
Many applied max/min problems involve finding an extreme value of a function, subject to a constraint . Often this can be done, as we have, by explicitly combining the equations and then finding critical points. There is another approach that is often convenient, the method of Lagrange multipliers.
• 14.E: Partial Differentiation (Exercises)
These are homework exercises to accompany David Guichard's "General Calculus" Textmap.

Thumbnail: A graph of $$x^2+xy+y^2=z$$ and $$y=1$$. We want to find the partial derivative at that leaves constant; the corresponding tangent line is parallel to the -axis. Image used with permission (CC BY-SA 3.0; Indeed123).