1. Section 15.5
The Chain Rule

2. 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

3. 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

4. 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.

5. 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.

6. 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

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