§2.3 Techniques of Differentiation

The derivative of a product of function is not the product of separate derivative!!

Suppose we have two function f(x)=x3 and g(x)=x6

The Product Rule If the two functions f(x) and g(x)

are differentiable at x, then so is the product

P(x)=f(x)g(x) and

§2.3 Techniques of Differentiation

)]([)()]([)()]()([ xfdx

dxgxg

dx

dxfxgxf

dx

d

''')( gffgfg

§2.3 Techniques of Differentiation

The Quotient Rule If the two functions f(x) and g(x) are differentiable at x, then so is the quotient Q(x)=f(x)/g(x) and

0)( if )(

)]([)()]([)(]

)(

)([

2

xg

xg

xgdx

dxfxf

dx

dxg

xg

xf

dx

d

2

''')(

g

fggf

g

f

a. Express the revenue R(t) for this product as a function of timeb. At what rate is revenue changing with respect to time after4 months? Is revenue increasing or decreasing at this time?

Solution: The revenue is given by

Example: A manufacturer determines that t months after a new product is introduced to the market, hundred units can be produced and then sold at a price of dollars per unit

tttx 3)( 2 302)( 2/3 ttp

)302)(3()()()( 2/32 ttttptxtR

]32)[302()]2

3(2)[3(

]3[)302(]302[)3()(

2/32/12

22/32/32

ttttt

ttdt

dtt

dt

dtttR

14)4( R Decreasing

Example: A biologist models the effect of introducing a toxin to a bacterial colony by the function

Where p is the population of colony (in millions) t hours after the toxin is introduced

a. At what rate is the population changing when the toxin is introduced? Is the population increasing or decreasing at this time?

b. At what time does the population begin to decrease?

4

1)(

2

tt

ttp

Solution: The rate of change of population with respect to time is given by

For 0≤t<1 P’(t)>0 and P(t) is increasing for t>1 P’(t)<0 and P(t) is decreasing

22

2

22

22

)4(

32t-

)4(

]4[)1(]1[)4()(

tt

t

tt

ttdtd

ttdtd

tttp

1875.016

3

)400(

)300()0(

2

p

The toxin is introduced when t=0, and at that time the population is changing at the rate

The population is decreasing when P’(t)<0. Since

)3)(1(322 tttt

22 )4(

3)1)(t-(t-)(

tt

tp

We can write

§2.3 The second Derivative

By differentiating this function, we get

This is a function and so it can be differentiated. Here is the notation that we’ll use for that as well as the derivative.

This is called the second derivative and is nowcalled the first derivative

Consider the following function

)(xf

Alternate Notation There is some alternate notation for higher order derivatives as well.

.

This is a function as so we can differentiate it again. This will be called the third derivative

Again,

§2.3 The Higher -order Derivative

Continuing,

§2.3 The Higher -order Derivative

The nth Derivative For any positive integer n, the nth derivative of a function is obtained from the function by differentiating successively n times. If the original function is y=f(x) the nth derivative is denoted by

)()( xf nn

n

dx

yd

xy

1

221 1

)(x

xxdx

d

dx

dy

332

2

2 22)(

xxx

dx

d

dx

yd

443

3

3 66)2(

xxx

dx

d

dx

yd

554

4

4 2424)6(

xxx

dx

d

dx

yd

§2.4 The Chain RuleSuppose the total manufacturing cost at a certain factory is a function of the number of units produced, which in turn is a function of the number of hours the factory has been operating. If C, q, t, denote the cost, units produced and time respectively, then

unit)per (dollars output torespect with

cost of change of rate

dq

dC

hour)per (units time torespect with

output of change of rate

dt

dq

hour)per (dollars dt

dq

dq

dC

dt

dC

The product of these two rates is the rate of change of cost with respect to time that is

§2.4 The Chain Rule (IMPORTANT)

The Chain Rule If y=f(u) is a differentiable function of u and u=g(x) is in turn a differentiable of x, then thecomposite function y=f(g(x)) is a differentiable functionof x whose derivative is given by the product

dx

du

du

dy

dx

dy

)())(( xgxgfdx

dy

or, equivalently, by

dx

dy1)2(3)2( 2232 xxy

uudu

dy63 2 and x

dx

du2

Example : Find if

Solution:

22 xu13 23 uuyNote that , where

)2)(63( 2 xuudx

du

du

dy

dx

dy

And according to the chain rule,

)(6

)2)](2(6)2(3[

)2)(63(

223

222

2

xx

xxx

xuudx

dy

Replace u with x2+2

§2.4 The Chain RuleLet’s look at the functions

then we can write the function as a composition.

differentiate a composition function using the Chain Rule.

The derivative is then

In general, We differentiate the outside function leaving the

inside function alone and multiply all of this by the derivative of

the inside function,

§2.4 The General Power RuleThe General Power Rule For any real number n and differentiable function h.

)]([)]([)]([ 1 xhdx

dxhnxh

dx

d nn

Think of [h(x)]n as the composite function nn uxhgxh g where )]([)]([

)]([)( and )( 1 xhdx

dxhnuug n

By the chain rule

)]([)]([)()]([)]([)]([ 1 xhdx

dxhnxhxhgxhg

dx

dxh

dx

d nn

Example: An environmental study of a certain suburban community suggests that the average daily level of carbon monoxide in the air will be parts per million when the population is p thousand. It is estimated that t years from now, the population of the community will be thousand. At what rate will the carbon monoxide level be changing with respect to time 3 years from now?

175.0)( 2 ppc

21.01.3)( ttp

Solution: The goal is to find when t=3. Since dt

dc

2/122/12 )175.0(2

1)]2(5.0[)175.0(

2

1 ppdp

dc

tdt

dp2.0and

175.0

1.0)2.0()175.0(

2

12

2/12

p

pttp

dt

dp

dp

dc

dt

dc

It follows from the chain rule that

yearper millionper parts 24.017)4(5.0

)3)(4(1.02

dt

dc

When t=3, p(3)=3.1+0.1(3)2=4, and so

§2.5 Marginal Analysis

假如用 П 表示厂商的利润，则 П ＝ TR － TC 。那么利润最大化就可以用 П ＝ TR － TC 的最大化来表示。

边际效益是经济学中的一个概念，它大体可以这样理解 :即一个市场中的经济实体为追求最大的利润，多次进行扩大生产，每一次投资所产生的效益都会与上一次投资产生的效益之间要有一个差，这个差就是边际效益。

Example: 你肚子很饿了，你只有钱可以买 5 个馒头吃。

第一个馒头的边际效益最大 , 因为你那时候最饿，最需要，你多花一点钱也愿意买； 第二个的边际效益就递减了，因为有 1 个馒头进肚了 ... 不是那么饿了。

第五个的边际效益最小，因为那个时候你几乎已经快饱了，馒头如果卖的贵的话，你一定不会买了。

每支出 1 个馒头的价钱产生的效益，也就是你感觉花钱买来的价值。从第一个向最后一个递减！这就是边际效益了。

什么时候总收益和总成本之差最大呢？答案是当边际收益等于边际成本时。

§2.5 Marginal AnalysisSuppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know how much it costs to produce one more unit of this product.

Example 1: Suppose the total cost in dollars per week by ABC Corporation for producing its best-selling product is given by Find the actual cost of producing the 101st item

The cost of producing the (x + 1)st item can be found by computing the average rate of change, that is by computing :

7.19391/)2090007.210939(100101

)100()101(

cc

Note that where x = 100 and h = 1

This will give us the actual cost of producing the next item. However, it is often inconvenient to use. For this reason, Itis usually approximated by the instantaneous rate of change of the total cost function evaluated at the specific point of interest.

20006.0)( 0 xxC 1940)100( C

So, we’ll define the marginal cost function as the derivative of the total cost function.

h

xChxCxC

xCxCxCxC

h

)()(lim)(

1

)()1()()1( 00

00

0000

In economics, the use of the derivative to approximatethe change in a quantity that results from a 1-unit increase in production is called marginal analysis

7.1939100101

)100()101(

cc

§2.5 Marginal Cost Marginal Cost: If C(x) is the total cost of producing x units of a commodity. Then the marginal cost of producing x0 units is thederivative , which approximates the additional cost incurred when the level of production is increased by one unit, from x0 to x0+1

)( 0xC )()1( 00 xCxC

§2.5 Marginal AnalysisMarginal Revenue and Marginal Profit: Suppose R(x) is the revenue generated when x unitsof a particular commodity are produced, and P(x) is the corresponding profit. When x=x0 units are being produced, then:

)( 0xP

)()1( 00 xPxP The marginal profit is , it approximates , the additional profit obtained by producing one more unit

)( 0xR)()1( 00 xRxR

The marginal revenue is , it approximates , the additional revenue generated by producing one more unit.

Example: A manufacturer estimates that when x units of a particular commodity are produced, the total cost will be

983

8

1)( 2 xxxC

)75(3

1)( xxp

dollars, and furthermore, that all x units

will be sold when the Price is dollars per unit

a. Find the marginal cost and the marginal revenue.

The marginal cost is . 34

1)( xxC

R(x)=(number of units sold )(price per unit)

2

3

125)]75(

3

1[)( xxxxxxp

The marginal revenue is

xxR3

225)(

b. Use marginal cost to estimate the cost of producing the ninth unit.

The cost of producing the ninth units is the change in cost as x increases from 8 to 9 and can be estimated by the marginal cost

5$3)8(4

1)8( C

c. What is the actual cost of producing the ninth unit?

13.5$)8()9( cC

d. Use marginal revenue to estimate the revenue derived from the sale of the ninth unit

67.19$)8(3

225)8( R

e. What is the actual revenue derived from the sale of the ninth unit?

33.19$)8()9( RR

§2.5 Marginal Analysis (Review)

Marginal Cost: If C(x) is the total cost of producing x units of a commodity. Then the marginal cost of producing x0 units is thederivative , which approximates the additional cost incurred when the level of production is increased by one unit, from x0 to x0+1

)( 0xC )()1( 00 xCxC

h

xChxCxC

xCxCxCxC

h

)()(lim)(

1

)()1()()1( 00

00

0000

Marginal analysis is an important example of a general Incremental approximation procedure

§2.5 Approximation by increments

§2.5 Approximation by increments

Approximation by Increment If f(x) is differentiable at x=x0 and x is a small change in x, then △

xxfxfxxf )()()( 000

)()( 00 xfxxff

xxff )( 0

Or, equivalently, if , then

Solution: A sphere of radius R and diameter x=2R has volume

Example: During a medical procedure, the size of a roughly tumor is estimated by measuring its diameter and using the

formula to compute its volume. If the diameter is measured as 2.5 cm with a maximum error of 2%, how accurate is the volume measurement?

3

3

4RV

33333 cm 181.8)5.2(6

1

6

1)

2(

3

4

3

4 x

xRV

xVVxVV )5.2()5.2()5.2(

to be continued

))5.2(02.0)](5.2([

)]5.2([Vvolume inerror Maximum

V

xV

817.9)5.2(2

1)5.2(

2

1)3(

6

1)( 222 VxxxV

491.0)05.0)(817.9(volume inerror Maximum

672.8690.7 V

§2.5 Approximation of percentage Change

The percentage change of a quantity expresses the change in that quantity as a percentage of its size prior to the change. In particular,

quantity of size

quantity in change100change of Percentage

§2.5 DifferentialsDifferentials The differential of x is dx= x, and if △y=f(x) is a differentiable function of x then is the differential of y

dxxfdy )(

§2.6 Implicit Differentiation Explicit form: y=f(x)

800020)( 2 xxxf32

1)(

3

x

xxf

Implicit form

xyyx 332 56 yxyyx 232 32

Implicit Differentiation Suppose an equation defines y implicitly

as a differentiable function of x. To find dx

dy

1. Differentiate both sides of equation with respect to x. remember that y is really a function of x and use the chain rulewhen differentiating terms containing y.

2. Solve the differentiated equation algebraically for dx

dy

322 ))(()( xxfxfx ][]))(()([ 322 xdx

dxfxfx

dx

d

)(

2

]))([()]([

22

3

22

3)(2)]()([x

dx

d

xfdx

dxfx

dx

d

xdx

dfxfx

dx

dxf

dx

dfx

)(2

)(232

2

xfx

xxfx

dx

df

22 3)(2)2)(( xdx

dfxfxxf

dx

dfx )(23)](2[ 22 xxfx

dx

dfxfx

Example: Find if 322 xyyx dx

dy

Solution: Differentiate both sides of the equation with respect to x. Don’t forget that y is actually a function of x.

yx

xyx

dx

dy

2

232

2

§2.6 Computing the slope of a Tangent Line by Implicit differentiation

Find the slope of the tangent line to the circle at the point (3,4). What is the slope at the point (3,-4)?

2522 yx

4

3

43)4,3(

yxy

x

dx

dy

Similarly, at (3,-4)

022 dx

dyyx

y

x

dx

dy

Solution: Differentiating both sides of the equation with respect to x

4

3

43)4,3(

yxy

x

dx

dy

The slope at (3,4) is the value of when x=3 and y=4

dx

dy

§2.6 Related RatesExample: The manager of a company determines that when q hundred units of a particular commodity are produced, the total cost of production is C thousand dollars, where . When 1500 units are being produced, the level of production is increasing at the rate of 20 units per week. What is the total cost at this time and at what rate is it Changing?

42753 32 qC

Both C and q can be regarded as function of a third variable t

This kind of problem is said to involve related rates.

C and q are related by equation 42753 32 qC

Implicit differentiation can be used to relate to dt

dCdt

dq

Solution:

Differentiating implicitly in the equation with respect to time, we get

42753 32 qC

dt

dCWe want to find when q=15 (1500 units) and )100/20(2.0

dt

dq

0]3[32 2 dt

dqq

dt

dCC

dt

dqq

dt

dCC 292

dt

dq

C

q

dt

dC

2

9 2

When q=15, the cost C satisfies

4275)15(3 32 C C=120

Substituting q=15,c=120 and into the formula for we obtain dt

dC0.2dqdt

6875.1)2.0]()120(2

)15(9[

2

dt

dC

Example 2. A lake is polluted by waste form a plant located on its shore. Ecologist determine that when the level of pollutant is x parts per million (ppn). There will be F fish of a certain species in the lake. Where

When there are 4000 fish left in the lake, the pollution is increasing at the rate of 1.4 ppm/year. At what rate is the fish population changing at this time

§2.6 Related Rates

dF

dtSolution: We want to find when F=4000 and When there are 4000 fish in the lake, the level of pollution x satisfies

1.4dx

dt

x=25

According to the chain rule

Substituting F=4000, x=25 and , we obtain 1.4dxdt

2

16000[ ](1.4) 70

25(3 25)

dF

dt

Summary Definition of the Derivative

h

xfhxfxf

h

)()(lim)(

0

Interpretation of the Derivative

Slope as a Derivative : The slope of the tangent line to the curve y=f(x) at point (c,f(c)) is )(tan cfm

Instantaneous Rate of Change as a Derivative: The rate of change of f(x) with respect to x when x=c is given by )(cf

Summary Sign of The Derivative )(xf

If the function f is differentiable at x=c, then

f is decreasing at x=c if <0)(cf f is increasing at x=c if >0)(cf

Techniques of Differentiation

0][ cdx

d 1][ nn nxxdx

d )]([)]([ xfdx

dcxcf

dx

d

)]([)]([)]()([ xgdx

dxf

dx

dxgxf

dx

d

)]([)()]([)()]()([ xfdx

dxgxg

dx

dxfxgxf

dx

d

0)( if )(

)]([)()]([)(]

)(

)([

2

xg

xg

xgdx

dxfxf

dx

dxg

xg

xf

dx

dThe Quotient Rule

The Product Rule

Summary

Application of Derivative

dx

du

du

dy

dx

dy

Tangent line, Rectilinear Motion, Projectile Motion

The Higher -order Derivative

The Chain Rule

)())(( xgxgfdx

dy

)]([)]([)]([ 1 xhdx

dxhnxh

dx

d nn The General Power Rule

Marginal Analysis and Approximation by increments

Summary

)( 0xC )()1( 00 xCxC The marginal cost is , it approximates ,the additional cost generated by producing one more unit.

h

xChxCxC

xCxCxCxC

h

)()(lim)(

1

)()1()()1( 00

00

0000

Approximation by Incrementxxfxfxxf )()()( 000

Implicit Differentiation and Related Rate