Differentiation 3 Basic Rules of Differentiation The Product and Quotient Rules The Chain Rule Marginal Functions in Economics Higher-Order Derivatives Implicit Differentiation and Related Rates Differentials

The Derivative

Basic Differentiation Rules 1. Ex. 2. Ex.

Basic Differentiation Rules 3. Ex. 4. Ex.

Basic Differentiation Rules APPLIED EXAMPLE 7 Conservation of a Species A group of marine biologists at the Neptune Institute of Oceanography recommended that a series of conservation measures be carried out over the next decade to save a certain species of whale from extinction. After implementing the conservation measures, the population of this species is expected to be N(t) ＝ 3t 3 ＋ 2t 2 － 10t ＋ 600 (0≤ t≤ 10) where N(t) denotes the population at the end of year t. Find the rate of growth of the whale population when t ＝ 2 and t ＝ 6 （ use definition of derivative ）. How large will the whale population be 8 years after implementing the conservation measures?

More Differentiation Rules 5. Ex. Product Rule Derivative of the first function Derivative of the second function

More Differentiation Rules 6.Quotient Rule Sometimes remembered as:

More Differentiation Rules 6. Ex. Quotient Rule (cont.) Derivative of the numerator Derivative of the denominator

More Differentiation Rules APPLIED EXAMPLE 6 Rate of Change of DVD Sales The sales (in millions of dollars) of a DVD recording of a hit movie t years from the date of release is given by Find the rate at which the sales are changing at time t. How fast are the sales changing at the time the DVDs are released (t ＝ 0)? Two years from the date of release?

The Chain Rule

Note: h(x) is a composite function. Another Version: The Chain Rule The Derivative is:

The Chain Rule leads to the General Power Rule: Ex. The Chain Rule

Chain Rule: Example Ex.

Chain Rule: Example Ex. Sub in for u

APPLIED EXAMPLE 7 Growth in a Health Club Membership The membership of The Fitness Center, which opened a few years ago, is approximated by the function N(t) ＝ 100(64 ＋ 4t) 2/3 (0 ≤ t ≤ 52) where N(t) gives the number of members at the beginning of week t. a. Find N’(t). b. How fast was the center’s membership increasing initially (t ＝ 0)? c. How fast was the membership increasing at the beginning of the 40th week? d. What was the membership when the center first opened? At the beginning of the 40th week? Chain Rule: Application

The profit P of a one-product software manufacturer depends on the number of units of its products sold. The manufacturer estimates that it will sell x units of its product per week. Suppose P ＝ g(x) and x ＝ f (t), where g and f are differentiable functions. 1.Write an expression giving the rate of change of the profit with respect to the number of units sold. 2.Write an expression giving the rate of change of the number of units sold per week. 3.Write an expression giving the rate of change of the profit per week. Chain Rule: Explore & Discuss

Suppose the population P of a certain bacteria culture is given by P ＝ f (T), where T is the temperature of the medium. Further, suppose the temperature T is a function of time t in seconds—that is, T ＝ g(t). Give an interpretation of each of the following quantities: Chain Rule: Explore & Discuss

Suppose c(x) is the total cost of producing x units of a certain commodity in a company. The function c(x) is called a cost function. The marginal cost function is the derivative, c’(x), of the cost function. Economists defined the marginal cost at a production level x to be c(x+1)-c(x), which is the cost of producing one additional unit of the commodity. Since it follows that the marginal cost c’(x) at the production level x is approximately the cost of producing the (x+1)st unit. Marginal Functions

The Marginal Cost Function approximates the change in the actual cost of producing an additional unit. The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit. The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.

Marginal Cost The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a certain level of operation Exercise: Interpret the value C'(200) ＝ 20

Cost Functions Given a cost function C(x) ＝ x 3 － 0.08x 2 ＋ 40x ＋ 5000, the Marginal Cost Function (MC) is the Average Cost Function (AC) is Group Discussion: Discuss and Sketch the graph of MC, AC for the given cost fuction in the same coordinate system.

Cost Functions Note: The graph of MC, AC will intersect at the lowest point of AC. Why?

Revenue Functions Given a revenue function, R(x), the Marginal Revenue Function (MR) is Question 1: What's the meaning of MR? Question 2: What's the mathematic expression of MR?

Revenue Functions Given a revenue function, R(x), the Marginal Revenue Function (MR) is

Revenue Functions Example: Suppose the demand function for the product is a. Find the revenue function R. b. Find the marginal revenue function R'. c. Compute R'(2000) and interpret your result.

Given a profit function, P(x), the Marginal Profit Function (MP) is Profit Functions

Exercise The monthly demand for T-shirts is given by where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is 1. Find the revenue and profit functions. 3. Find the marginal average cost function. …… 2. Find the marginal cost, marginal revenue, and marginal profit functions.

Solution 1. Find the revenue and profit functions. 2. Find the marginal cost, marginal revenue, and marginal profit functions. Revenue = xp Profit = revenue – cost Marginal Cost =

Solution 3. Find the marginal average cost function. 2. (cont.) Find the marginal revenue and marginal profit functions. Marginal revenue = Marginal profit =

Elasticity of Demand

 Price elasticity is a measure of the responsiveness of quantity demanded to changes in prices.  Definition: It is defined as the percentage change in the quantity demanded relative to the percentage change in price, when moving from one point to another on a demand curve. Elasticity of Demand

If f is a differentiable demand function defined by Then the elasticity of demand at price p is given by Demand is:Elastic if E(p) > 1 Unitary if E(p) = 1 Inelastic if E(p) < 1

Example Consider the demand equation which describes the relationship between the unit price p in dollars and the quantity demanded x of the Acrosonic model F loudspeaker systems. a. Find the elasticity of demand b. Compute E(300) and interpret your result.

Example Interpret: When the price is set at $300 per unit, an increase of 1% in the unit price will cause a decrease of approximately 3% in the quantity demanded.

Summary: Price Elasticity of Demand and Change in Revenue _ If price  If price  ELASTIC Revenue decrease Revenue increase INELASTIC Revenue increase Revenue decrease UNITARY No change No change

Homework 1. (In-class) Interpret relationship between Price Elasticity of Demand and Change in Revenue. 2. (Online) Classify and sketch each category of elasticity of demand and submit your homework on the course website.

Higher Derivatives The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative. DerivativeNotations Second Third Fourth nth

Example of Higher Derivatives Givenfind

Example of Higher Derivatives

Givenfind

Application of Higher Derivatives  acceleration  curb the property price hikes  Inflation is moderating

Implicit Differentiation  Form 1 ：  Form 2 ： Question: What's difference between two forms?

Implicit Differentiation y is expressed explicitly as a function of x. y is expressed implicitly as a function of x.

Implicit Differentiation To differentiate the implicit equation, we write f (x) in place of y to get: It is not necessary to solve an equation for y in terms of x in order to find the derivative of y. Instead we can use the method of implicit differentiation, in which we differentiate each term of the equation with respect to x and then solve the resulting equation for y’. Example 1

Implicit Differentiation (cont.) Now differentiate using the chain rule: which can be written in the form subbing in y Solve for y’

Example Find dy/dx if Solution: Solution: We are going to differentiate both sides of the given equation w.r.t. x. Firstly, we temporarily replace y by f(x) and rewrite the equation as. Secondly, we differentiate both sides of this equation term by term w.r.t.x: We are going to differentiate both sides of the given equation w.r.t. x. Firstly, we temporarily replace y by f(x) and rewrite the equation as. Secondly, we differentiate both sides of this equation term by term w.r.t. x:

Thus, we have Finally, replace f(x) by y to get

Implicit Differentiation: Suppose an equation defines Implicit Differentiation: Suppose an equation defines y implicitly as a differentiable function of x. To find y implicitly as a differentiable function of x. To find df/dx df/dx : 1. Differentiate both sides of equation w.r.t. x. remember that y is really a function of x and use the remember that y is really a function of x and use the chain rule when differentiating terms containing y. chain rule when differentiating terms containing y. 2. Solve the differentiated equation algebraically for dy/dx. dy/dx. Exercise Find dy/dx by implicit differentiation where

Thus, the slope at (3,4) is The slope at (3,-4) is Differentiating both sides of the equation w.r.t. x, we have Find the slope of the tangent line to the circle at the point (3,4). What is the slope at the point (3,-4)? Application 1: Computing the slope of a tangent line by implicit differentiation Example Solution: Solution:

Application to Economics Application 2: Application to Economics Example Suppose the output at a certain factory is units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 30 hours of skilled labor and 20 hours of unskilled labor. Question: Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level.

Solution: Solution: If output is to be maintained at the current level, which is the value of Q when x=30 and y=20, the relationship between skilled labor x and unskilled labor y is given by the equation The goal is to estimate the change in y that corresponds to a 1-unit increase in x when x and y are related by above equation. As we know, the change in y caused by a 1-unit increase in x can be approximated by the derivative dy/dx. Using implicit differentiation, we have Now evaluate this derivative when x=30 and y=20 to conclude that

Related Rates Look at how the rate of change of one quantity is related to the rate of change of another quantity. Ex.Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? Note: The rate of change of the distance between them is related to the rate at which the cars are traveling.

Related Rates Steps to solve a related rate problem: 1. Assign a variable to each quantity. Draw a diagram if appropriate. 2. Write down the known values/rates. 3. Relate variables with an equation. 4. Differentiate the equation implicitly. 5. Plug in values and solve.

Ex. Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? Distance = z x y From original relationship

Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm 3 /s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Solution Let V be the volume of the balloon and let r be its radius. In order to connect dV/dt and dr/dt, we first relate V and r by the formula for the volume of a sphere:

We differentiate each of this equation with respect to t: Now we solve for the unknown quantity: If we put r = 25 and dV/dt = 100 in this equation, we get The radius of the balloon is increasing at the rate of cm/s.

3.7 Differentials-Example Butterfly effect: a hurricane being influenced by minor perturbations such as the flapping of the wings of a distant butterfly several weeks earlier. A very small change in initial conditions had created a significantly different outcome.

3.7 Differentials - Example The People's Bank of China (PBOC), China's central bank, cut the benchmark rate for one-year deposits by 25 basis points and the one-year lending rate by 40 basis points from Saturday.

3.7 Differentials-Example Introduction: Side of the square increase from to what’s the change of its area?

Increments( 增量 ) An increment in x represents a change from x 1 to x 2 and is defined by: Read “delta x” An increment in y represents a change in y and is defined by:

M N T ） Determine the meaning of the following quantities: Δx Δy dx dy P

Differentials Let y = f (x) be a differentiable function, then the differential of x, denoted dx, is such that The differential of y, denoted dy, is Note: x is the initial value

Rules of Differentials

Derivative Rule Differential Rule

Example 1 Given

Example 2 Given the implicit equation xy ＋ y ＝ 1, find dy.

Example 3 Linear Approximations Approximate the following value using differentials:

Example The total cost incurred in operating a certain type of truck on a 500-mile trip, traveling at an average speed of v mph, is estimated to be Find the approximate change in the total operating cost when the average speed is increased from 55 mph to 58 mph.