1 §4.2 The Chain Rule (Pages 251~261) Composite Functions: The Chain Rule: Lectures 9 & 10

2 Define A Composite Function And say that h is the composite of f and u. We read f(u(x)) as “f of u of x”

3 Generalized Power Rule Ex. 1

4 Generalized Power Rule Example Ex. 3

5 The Chain Rule The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity. If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and

6 Chain Rule Examples Original Rule Generalized Rule Power Rule Generalized Power Rule

7 Why shall we accept the chain rule? And so on for higher positive power of u. Similarly, we can use quotient rule to verify the chain rule for negative powers.

8 Chain Rule Example Ex. 4

9 Chain Rule Example Ex. 5

10 Chain Rule Example Ex. 6

11 Chain Rule Example Ex. 7

12 Harder Examples Using the Chain Rule Ex. 8

13 Harder Examples Using the Chain Rule Ex. 9

14 Chain Rule in Differential Notation If y is a differentiable function of u and u is a differentiable function of x, then If y = u 3, where u = 4x + 1, then Quick Example

15 Chain Rule Example Ex. 10 Sub in for u

16 Ex. 11 Marginal ProductPrecision Manufacturer is informed by a consultant that its annual profit is given by P =  200, q – 0.46q 2 – q 3, where q is the number of surgical lasers it sells each year. The consultant also informs Precision that the number of surgical lasers it can manufacture each year depends on the number n of assembly-line workers it employs according to the equation q = 100n (i.e., each worker contributes 100 lasers per year). Use the chain rule to find the marginal product dP/dn. Chain Rule Example

17 Solution:

18 Ex. 12 Marginal RevenueSuppose a company’s weekly revenue R is given as a function of the unit price p, and p in turn is given as a function of weekly sales q by means of a demand equation). If and find the marginal revenue when sales are 1000 items per week. Chain Rule Example

19 Solution: Units: revenue per item = revenue per $1 price increase  price increase per additional item Thus, if the price is lowered to increase the demand from 1000 to 1001 items per week, the weekly revenue will drop by approximately $800.

20 Hw Problem 56. Online TradingThe profitability p (measured in quarterly net income) of the Charles Schwab Corporation increased as more of its customers traded online according to the formula During that time, the fraction of online trades increased according to Use direct substitution to express the quarterly profits p as a function of time (do not simplify the expression) and then use the chain rule to estimate the rate of change of profitability at the beginning of September Be sure to specify the units.

21 Hints: Use direct substitution, we have Then, Compute

PollutionAn offshore oil well is leaking oil and creating a circular oil slick. If the radius of the slick is growing at a rate of 2 miles/hour, find the rate at which the area is increasing when the radius is 3 miles. (The area of a disc of radius r is A =  r 2.) Chain Rule Example Solution: (  HW Problem 66)