Section 15.5 The Chain Rule

THE CHAIN RULE (CASE 1) Suppose that z = f (x, y) is a differentiable function of x and y, where x = g(t) and y = h(t) are both differentiable functions of t. Then z is a differentiable function of t and

THE CHAIN RULE (CASE 2) Suppose that z = f (x, y) is a differentiable function of x and y, where x = g(s, t) and y = h(s, t) are differentiable functions of s and t. Then

THREE TYPES OF VARIABLES In Case 2 of the Chain Rule, there are three types of variables The variables s and t are independent variables. The variables x and y are intermediate variables. The variable z is the dependent variable.

THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2, . . . , xn and each xj is a differentiable function of the m variables t1, t2, . . . , tm. Then u is a function of t1, t2, . . . , tm and for each i = 1, 2, . . . , m.

IMPLICIT DIFFERENTIATION Suppose that F(x, y) = 0 defines y implicitly as a function of x. If we want to find dy/dx, we can use the techniques from Calculus I (Section 3.7). However, using the Chain Rule on both sides of F(x, y) = 0 and solving for dy/dx, we get the formula

IMPLICIT DIFFERENTIATION (CONTINUED) If z is an implicit function of x and y defined by the equation F(x, y, z) = 0, we have