1. MAT 213 Brief Calculus Section 3.4 The Chain Rule

2. Consider… An investor has been buying gold at a constant rate of 0.2 ounces per day. If gold is currently worth $323.10 per ounce, how quickly is the total value of the investor’s gold increasing per day? How do we calculate this? How could we use our derivative notation? Let g = gold, o = ounce, and t = time (measured in days What are the units?

3. Consider… A car is traveling at 100 feet per second. What is the car’s speed in miles per hour? Hint: There are 5280 feet in a mile How can we calculate this? How can we write this in derivative notation? Let f = feet, m = mile, s = second, and h = hour

4. If y is a function of u and u is a function of x, then… THE CHAIN RULE (Form 1)

5. Recall… Composition of Functions Results (outputs) of one process are used as inputs for another process. NOTATION Take the functions f(x) and g(x) f(g(x)) = (f◦g)(x) We can also have composition of more than two functions f(g(h(k(x))))

6. Recall… Composition of Functions So we can think of composite functions as functions that have functions inside of them.

7. Composition of Functions E XAMPLES The function ln(x 2 ) can be thought of as a composition of two other functions, ln x and x 2, with the x 2 being INSIDE the ln function.

8. Consider the function –We can “decompose” this function into two functions we know how to take the derivative of –For example –What are –Now think of as

9. The Derivative of a Composite Function Let u=g(x) and y=f(u) be differentiable functions So y=f(g(x)) is also differentiable THE CHAIN RULE (Form 2) The derivative of a composite function is the derivative of the outside function (leaving the inside function alone) times the derivative of the inside function

10. The Chain Rule… In Words Derivative of the outside function with the inside untouched Derivative of the inside function Derivative of a composite function = x

11. The Chain Rule allows us to generalize some of our previous rules “Simple” Rules Generalized (Chain) Rules

12. The Chain Rule Examples