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Chain Rule
The Chain Rule is more general than the Inverse Function Rule and deals with the case where x and y are both functions of a third variable t. Suppose x = f(t), y = G(x). Thus x depends on t, and y depends on x. But y is also a function of t, y = g(t), where g is defined by the rule g(t) = G(f (t)). The function g is sometimes called the composition of G and f (sometimes written g = G ◦ f). The composition of G and f may be described in terms of black boxes. The function g = G ◦ f is a large black box operating on the input t to produce g(t) = G(f(t)). If we look inside this black box (pictured in Figure 2.6.1), we see two smaller black boxes, f and G. First f operates on the input t to produce f(t), and then G operates on f(t) to produce the final output g(t) = G(f(t)). The Chain Rule expresses the derivative of g in terms of the derivatives of f and G. It leads to the powerful method of "change of variables" in computing derivatives and, later on, integrals. Figure 2.6.1


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