10: Derivatives of Multivariable Functions

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10.1: Limits
In this section, we will study limits of functions of several variables, with a focus on limits of functions of two variables. In single variable calculus, we studied the notion of limit, which turned out to be a critical concept that formed the basis for the derivative and the definite integral. In this section we will begin to understand how the concept of limit for functions of two variables is similar to what we encountered for functions of a single variable.
10.2: First-Order Partial Derivatives
Now that we are investigating functions of two or more variables, we can still ask how fast the function is changing, though we have to be careful about what we mean. Thinking graphically again, we can try to measure how steep the graph of the function is in a particular direction. Alternatively, we may want to know how fast a function's output changes in response to a change in one of the inputs. Over the next few sections, we will develop tools for addressing issues such as these.
10.3: Second-Order Partial Derivatives
In what follows, we begin exploring the four different second-order partial derivatives of a function of two variables and seek to understand what these various derivatives tell us about the function's behavior.
10.4: Linearization- Tangent Planes and Differentials
One of the central concepts in single variable calculus is that the graph of a differentiable function, when viewed on a very small scale, looks like a line. We call this line the tangent line and measure its slope with the derivative. In this section, we will extend this concept to functions of several variables.
10.5: The Chain Rule
In the case of a function f of two variables where z=f(x,y), it might be that both x and y depend on another variable t. A change in t then produces changes in both x and y, which then cause z to change. In this section we will see how to find the change in z that is caused by a change in t, leading us to multivariable versions of the Chain Rule involving both regular and partial derivatives.
10.6: Directional Derivatives and the Gradient
It is natural to wonder how we can measure the rate at which a function changes in directions other than parallel to a coordinate axes. In what follows, we investigate this question, and see how the rate of change in any given direction is connected to the rates of change given by the standard partial derivatives.
10.7: Optimization
In multivariable calculus, we are often interested in finding the greatest and/or least value(s) that a function may achieve. Moreover, there are many applied settings in which a quantity of interest depends on several different variables. In the following preview activity, we begin to see how some key ideas in multivariable calculus can help us answer such questions by thinking about the geometry of the surface generated by a function of two variables.
10.8: Constrained Optimization - Lagrange Multipliers
Some optimization problems involve maximizing or minimizing a quantity subject to an external constraint. In these cases the extreme values frequently won't occur at the points where the gradient is zero, but rather at other points that satisfy an important geometric condition. These problems are often called constrained optimization problems and can be solved with the method of Lagrange Multipliers, which we study in this section.