The Multi-Variable Chain Rule: Derivatives of Compositions
Suppose that f(x,y) depends on two variables but that the x(t) and y(t) are themselves both functions of t. Then f(x(t), y(t)) is a composition of functions and the derivative of f with respect to t is computed via the multi-variable Chain Rule. In fact, this scenario is one of many different generalizations of the single variable chain rule. We can use arrow (or dependency) diagrams to illustrate the relationships between a bunch of multivariable functions and for each situation write down a chain rule that gives the slope. The idea in the example given here is that a small change in t results in a small change in both x and y which in turn result in a small change in f.
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