Topic 4: Differentiation
Topic 5: Differentiation
Lecture Notes:
section 4
Jacques Text Book (edition 3):
Chapter4
Recall measuring change in the case of a linear function:
y = a + bx
a = intercept
b = slope i.e. the impact of a unit change in x on the level of y
b = [pic] = [pic]
( constant along a straight line
( y changes at a constant rate in response to changes in x
If the function is non-linear: e.g. if y = x2
[pic] = [pic] 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
- which is different for different values of x
y = x2
y+(y = (x+(x) 2
y+(y =x2+2x.(x+(x2
(y = x2+2x.(x+(x2 – y
since y = x2
(y = 2x.(x+(x2
[pic] = 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
[pic] = 2x in the limit, as (x (0
[pic]
Rules for Differentiation (section 4.3)
1. The Constant Rule
If y = c where c is a constant,
[pic]
e.g. y = 10 then [pic]
2. The Linear Function Rule
If y = a + bx
[pic]
e.g. y = 10 + 6x then [pic]
3. The Power Function Rule
If y = axn, a & n are constants
[pic]
i) y = 4x => [pic]
ii) y = 4x2 => [pic]
iii) y = 4x3 => [pic]
iv) y = 4x-2 => [pic]
4. The Sum-Difference Rule
If y = f(x) ( g(x)
[pic]
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
[pic]
(ii) y = 4x2 - x3 - 4x then [pic]
(iii) y = 5x + 4 then [pic]
5. The Product Rule
If y = u.v where u and v are functions of x
Then [pic]
i) y = (x+2)(ax2+bx)
[pic]
ii) y = (4x3-3x+2)(2x2+4x)
[pic]
6. The Quotient Rule
If y = u/v where u and v are functions of x
Then [pic]
i) y = (x+2)/(x+4)
[pic]
ii) y = (3x+2)/(x2+4)
[pic]
[pic]
7. The Chain Rule
If y is a function of v, and v is a function of x, then y is a function of x and
[pic]
i) y = (ax2 + bx)½
let v = (ax2 + bx) , so y = v½
[pic]
ii) y = (4x3 + 3x – 7 )4
let v = (4x3 + 3x – 7 ), so y = v4
[pic]
8. The Inverse Function Rule
If x = f(y) then [pic]
The derivative of the inverse of the function x = f(y), is the inverse of the derivative of the function
(i) x = 3y2 then
[pic] so [pic]
(ii) y = 4x3 then
[pic] so [pic]
1. Differentiating functions using Rules 1 ( 8,
See Section 4 of course manual, questions 3, 4 and 10
Applications of the Basic Rules
Calculating Marginal Functions
Example 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
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….
1) Find the Average Cost
AC = TC / Q = Q2 – 4Q + 12
2) Find the Marginal Cost
[pic]
3) When does AC = MC?
Q2 – 4Q + 12 = 3Q2 – 8Q + 12
( 2Q2 – 4Q = 0
( 2Q = 4
( Q = 2
Thus, AC = MC curves when Q = 2
4) When does the slope of AC = 0?
Differentiate AC = Q2 – 4Q + 12 to find slope……
[pic]
then set it equal to 0
2Q – 4 = 0
( 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
Example 3: ELASTICITY
Price Elasticity of Demand:
ed = [pic]
= [pic] = [pic]
To calculate the point elasticity of demand then,
ed = [pic]
e.g. Find ed of the function Q = aP-b
ed = [pic]
= [pic]
Inelastic demand: if (ed( < 1
Unit elastic demand: if (ed( = 1
Elastic demand: if (ed( > 1
9. Differentiating Exponential Functions (Course Manual, parts of Topic 6.1)
Aside: The exponential function:
y = exp(x) = ex
Features of y = ex
1. non-linear
2. always positive
3. as ( x get ( y and ( slope of graph
exponential function can be differentiated
Rule 9:
If y = ex then [pic] where e = 2.71828….
More generally,
If y = Aerx then [pic]
Examples:
1) y = e2x then [pic] = 2e2x using above rule
2) y = e-7x then [pic] = -7e-7x
.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
5. If y = ex then [pic] = y
6. From The Inverse Function Rule
y = ex ( [pic]
7. Now, if y = ex this is equivalent to writing x = loge y = ln y
8. Thus, x = ln y ([pic]
Rule 9: Differentiating Natural Logs
if y = loge x = ln x ( [pic]
NOTE: the derivative of a natural log function does not depend on the co-efficient of x
Thus, if y = ln mx ( [pic]
Proof
if y = ln mx m>0
Rules of Logs ( y = ln m+ ln x
Differentiating (Sum-Difference rule)
[pic]
Examples:
1) y = ln 5x (x>0) ( [pic]
2) y = ln(x2+2x+1)
let v = (x2+2x+1) so y = ln v
Chain Rule: ( [pic]
[pic]
[pic]
3) y = x4lnx
Product Rule: (
[pic]
= [pic] = [pic]
4) y = ln(x3(x+2)4)
Simplify first using rules of logs
( y = lnx3 + ln(x+2)4
( y = 3lnx + 4ln(x+2)
[pic]
Note:
2. Differentiating exponential and log functions using Rules 9 and 10,
See Section 6 of course manual, questions 3 and 4**
Example 1
If the Demand equation is given by
P = 200 – 40ln(Q+1)
Calculate the price elasticity of demand when Q = 20
Solution
Price elasticity of demand:
ed = [pic] ................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- week 1 calculus i practice problem solutions
- 1 compute the following derivatives simplify your
- section 14 5 3 23 08 directional derivatives and
- pol571 lecture notes expectation and functions of random
- common derivatives and integrals
- differentiating logarithm and exponential functions
- derivation of the inverse hyperbolic trig functions
- partial derivatives examples and a quick review of
- 5 numerical differentiation
- differentiation of exponential functions
Related searches
- broad topic narrowed topic activity
- logarithmic differentiation calculator
- differentiation of log log x
- ln differentiation rules
- log differentiation rules
- basic rules of differentiation pdf
- differentiation rules and problems
- introduction to differentiation pdf
- calculus differentiation rules pdf
- differentiation of transcendental functions
- parametric differentiation calculator
- differentiation of parametric equations