2.5. THE CHAIN RULE
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In this section, we encountered the following important ideas:
• A composite function is one where the input variable x first passes through one
function, and then the resulting output passes through another. For example, the
function h(x) = 2 sin(x) is composite since x −→ sin(x) −→ 2 sin(x) .
• Given a composite function C(x) = f (g(x)) that is built from differentiable functions
f and g, the chain rule tells us that we compute C ′ (x) in terms of f , g, f ′ , and g ′
according to the formula
C
′ (x) = f
′ (g(x))g
′ (x).
Exercises
1. Consider the basic functions f (x) = x 3 and g(x) = sin(x).
(a) Let h(x) = f (g(x)). Find the exact instantaneous rate of change of h at the
point where x =
π
4 .
(b) Which function is changing most rapidly at x = 0.25: h(x) = f (g(x)) or
r(x) = g( f (x))? Why?
(c) Let h(x) = f (g(x)) and r(x) = g( f (x)). Which of these functions has a
derivative that is periodic? Why?
2. Let u(x) be a differentiable function. For each of the following functions, determine the
derivative. Each response will involve u and/or u ′ .
(a) p(x) = e u(x)
(b) q(x) = u(e x )
(c) r(x) = cot(u(x))
(d) s(x) = u(cot(x))
(e) a(x) = u(x 4 )
(f) b(x) = u 4 (x)
3. Let functions p and q be the piecewise linear functions given by their respective graphs
in Figure 2.7. Use the graphs to answer the following questions.
(a) Let C(x) = p(q(x)). Determine C ′ (0) and C ′ (3).
(b) Find a value of x for which C ′ (x) does not exist. Explain your thinking.
(c) Let Y (x) = q(q(x)) and Z(x) = q(p(x)). Determine Y ′ (−2) and Z ′ (0).