Topic 5: Differentiation

Lecture Notes: section 4

Jacques Text Book (edition 3): Chapter4

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Recall measuring change in the case of a linear function:

y = a + bxa = interceptb = slope i.e. the impact of a unit change in x on the level of y

b = =

constant along a straight line y changes at a constant rate in response to changes in x

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If the function is non-linear: e.g. if y = x2

= gives slope of the line connecting 2 points (x1, y1) and (x2,y2) on a curve (2,4) to (4,16): slope = (16-4)/(4-2) = 6 (2,4) to (6,36): slope = (36-4)/(6-2) = 8

The slope of a curve is equal to the slope of the line (or tangent) that touches the curve at that point

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Total Cost Curve: y=x2

0

10

20

30

40

0 1 2 3 4 5 6X

y=x2

- which is different for different values of x

y = x2

y+y = (x+x) 2 y+y =x2+2x.x+x2

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Total Cost Curve

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7

X

y=x2

y = x2+2x.x+x2 – ysince y = x2

y = 2x.x+x2

= 2x+x

The slope depends on x and x

Differentiation: finds the derived function by letting change in x become arbitrarily small, i.e. letting x 0 = 2x in the limit, as x 0

Rules for Differentiation (section 4.3)

1. The Constant RuleIf y = c where c is a constant,

e.g. y = 10 then

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2. The Linear Function RuleIf y = a + bx

e.g. y = 10 + 6x then

3. The Power Function RuleIf y = axn, a & n are constants

i) y = 4x =>

ii) y = 4x2 =>

iii) y = 4x3 =>

iv) y = 4x-2 =>

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4. The Sum-Difference RuleIf y = f(x) g(x)

If y is the sum/difference of two or more functions of x: differentiate the 2 (or more) terms separately, then add/subtract(i) y = 2x2 + 3x then

(ii) y = 4x2 - x3 - 4x then (iii) y = 5x + 4 then

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5. The Product RuleIf y = u.v where u and v are functions of xThen

i) y = (x+2)(ax2+bx)

ii) y = (4x3-3x+2)(2x2+4x)

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6. The Quotient RuleIf y = u/v where u and v are functions of x

Then

i) y = (x+2)/(x+4)

ii) y = (3x+2)/(x2+4)

7. The Chain RuleIf y is a function of v, and v is a function of x, then y is a function of x and

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i) y = (ax2 + bx)½ let v = (ax2 + bx) , so y = v½

ii) y = (4x3 + 3x – 7 )4 let v = (4x3 + 3x – 7 ), so y = v4

8. The Inverse Function Rule

If x = f(y) then

The derivative of the inverse of the function x = f(y), is the inverse of the derivative of the function

(i) x = 3y2 then 10

so

(ii) y = 4x3 then so

- Differentiating functions using Rules 1 8,

See Section 4 of course manual, questions 3, 4 and 10

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Applications of the Basic Rules

Calculating Marginal FunctionsExample 1

A firm faces the demand curve P=17-3Q(i) Find an expression for TR in terms of Q(ii) Find an expression for MR in terms of Q

Solution:TR = P.Q = 17Q – 3Q2

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Example 2:

If a firms Total Cost Curve is: TC = Q3 – 4Q2 + 12Q

(i) Find an expression for AC in terms of Q

(ii)Find an expression for MC in terms of Q

(iii) When does AC=MC?

(iv) When does the slope of AC=0?

(v) Plot MC and AC curves and comment on the economic significance of their relationship(vi) Suppose now TC=Q3- 4Q2+12Q +10.

Draw new curves and comment….

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1) Find the Average CostAC = TC / Q = Q2 – 4Q + 12

2) Find the Marginal Cost

3) When does AC = MC?Q2 – 4Q + 12 = 3Q2 – 8Q + 12 2Q2 – 4Q = 0 2Q = 4 Q = 2Thus, AC = MC curves when Q = 24) When does the slope of AC = 0?Differentiate AC = Q2 – 4Q + 12 to find slope……

then set it equal to 02Q – 4 = 0

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Q = 2 when slope AC = 0

(v) Economic Significance? MC curve cuts the AC curve at its minimum point…….(draw both curves)MC cuts AC curve at minimum point…

(vi) What happens if we introduce Fixed costs to the TC function? TC=Q3- 4Q2+12Q +10 no impact on the MC function, shift up AC function by FC/q

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Example 3: ELASTICITYPrice Elasticity of Demand: ed =

= = To calculate the point elasticity of demand then,

ed = e.g. Find ed of the function Q = aP-b

ed = =

Inelastic demand: if ed < 1 Unit elastic demand: if ed = 1 Elastic demand: if ed > 1

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9. Differentiating Exponential Functions (Course Manual, parts of Topic 6.1)

Aside: The exponential function: y = exp(x) = ex

Features of y = ex non-linear always positive as x get y and slope of graph exponential function can be differentiated Rule 9: If y = ex then where e = 2.71828….

More generally,If y = Aerx then Examples:1) y = e2x then = 2e2x using above rule

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2) y = e-7x then = -7e-7x

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.Differentiating Natural Logs(Course Manual, Topic 6.2)

Thus, if y = ex then x = loge y = ln y

Logs to the base e are natural logs

Differentiating Natural Logs If y = ex then = y

From The Inverse Function Rule y = ex

Now, if y = ex this is equivalent to writing x = loge y = ln y

Thus, x = ln y

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Rule 9: Differentiating Natural Logs

if y = loge x = ln x

NOTE: the derivative of a natural log function does not depend on the co-efficient of xThus, if y = ln mx Proof

if y = ln mx m>0

Rules of Logs y = ln m+ ln x

Differentiating (Sum-Difference rule)

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Examples:1) y = ln 5x (x>0)

2) y = ln(x2+2x+1)

let v = (x2+2x+1) so y = ln vChain Rule:

3) y = x4lnx

Product Rule:

= =

4) y = ln(x3(x+2)4)

Simplify first using rules of logs

y = lnx3 + ln(x+2)4

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y = 3lnx + 4ln(x+2)

Note:

- Differentiating exponential and log functions using Rules 9 and 10,

See Section 6 of course manual, questions 3 and 4**

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Example 1If the Demand equation is given byP = 200 – 40ln(Q+1)Calculate the price elasticity of demand when Q = 20

Solution Price elasticity of demand: ed = <0 P is expressed in terms of Q, so find

Inverse rule of differentiation Thus, Hence, ed = <0The price elasticity of demand when Q is 20 is therefore computed as

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= -2.05

where P = 200 – 40ln(20+1) = 78.22

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Natural Logs in Applied Examples(see section 6.2 in Manual)

Useful in considering proportional changes in variables….

The derivative of log(f(x)) f’(x) / f(x), or the proportional change in the variable x

i.e. y = f(x), then the proportional x

= =

1) Show that if y = x, then

and this derivative of ln(y) with respect to x.

Solution:

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Now ln y = ln x

Re-writing ln y = lnx

Differentiating the ln y with respect to x gives the proportional change in x.Example :

If Price level at time t is P(t) = a+bt+ct2

The inflation rate at t is

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This is equivalent to differentiating the log of P(t) wrt t directly

lnP(t) = ln(a+bt+ct2)

where v = (a+bt+ct2) so lnP = ln v

Using chain rule,

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