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