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Basics of Differential Calculus
Professor Peter Cramton
Economics 300
Why differential calculus?
• Models explain economic behavior with system of
equations
• What happens if a variable changes?
– Comparative statics determines marginal change in
economic behavior
• How does change in tax rate alter consumption?
• How does change in NBA collective bargaining agreement
impact
– share of NBA revenues going to players?
– parity of teams across league?
Why differential calculus?
• Economic models assume rational optimizers
– Consumers maximize utility
– Producers maximize profits
– NBA owners maximize combination of wins and profits
• Optimization uses calculus to evaluate tradeoffs
– How much to consume?
• Consume until marginal utility = price
– How much to produce?
• Produce until marginal revenue = marginal cost
– Which free agents to go for?
Average rate of change over [x0,x1]
1 0
1 0
1 0 1 0
( ) ( ), where
and
f x f xy
x x x
x x x y y y
Average rate of change examples
1 0
1 0
( ) ( )f x f xy
x x x
1 0
1 0
: bx bxy
y a bx bx x x
2 22 1 0 1 0 1 0
1 0
1 0 1 0
( )( ):
x x x x x xyy x x x
x x x x x
6y
x
3y
x
2y x
Average rate of change
and difference quotient
1
1
( ) ( ) ( ) ( )y f x f x f x x f x
x x x x
2 2 22 ( ) 2
: 2y x x x x x x
y x x xx x x
( ):
y b x x bxy a bx b
x x
Some properties
• Rate of change of sum = sum of rates of change
– y, w, z are functions of x and y = w + z
– Then
• Scaling:
( )y w z w z
x x x x
( )ay ya
x x
Application: quadratic
2
2( ) 1
(2 )
y ax bx c
y x xa b c
x x x x
a x x b
2 2 2 2( ) ( ) 22
x x x x x x xx x
x x x
Application: cubic 3 2 2 3
3 2 2 3 3
2 2
( ) ( )( 2 )
( 3 3 )
3 3
x x x x x x x x
x x
x x x x x x x
x
x x x x
3 2
3 2
2 2
( ) ( ) 1
(3 3 ) (2 )
y gx ax bx c
y x x xg a b c
x x x x x
g x x x x a x x b
Exercise • Find difference quotient for each function
2
2
5
30 15
6 2 9
1
y x
y x
y x x
y x
5y
x
15y
x
6(2 ) 2y
x xx
(2 )y
x xx
Exercise
• Total revenue: TR = P Q
• Price: P = 10 .5Q
• Difference quotient?
• If Q = 5, what is impact of 1 unit increase in Q?
2(10 .5 ) .5 10
.5(2 ) 10
TR Q Q Q Q
TRQ Q
Q
.5(2(5) 1) 10 4.5TR
Q
Derivative is difference quotient as x0
0
( ) ( )limx
dy f x x f x
dx x
0
( ): lim
x
dy b x x bxy a bx b
dx x
2 22
0 0
( ): lim lim 2 2
x x
dy x x xy x x x x
dx x
3 2 2 2
0: lim 3 3 3
x
dyy x x x x x x
dx
Some properties
• Derivative of sum = sum of derivatives
– y, w, z are functions of x and y = w + z
– Then
• Scaling:
• Application
( )dy d w z dw dz
dx dx dx dx
( )d ay dya
dx dx
3 2
3 2
2
( ) ( ) 1
3 2
y gx ax bx c
dy d x d x dx dg a b c
dx dx dx dx dx
gx ax b
Derivative is difference quotient as x0
average rate of change difference quotient derivative
Derivative is rate of change as x0
Derivative is instantaneous rate of change
1
1
0
( ) ( ) ( ) ( )
( ) ( )limx
y f x f x f x x f x
x x x x
dy f x x f x
dx x
Tangent line is limit of secant line
Derivative is slope of tangent line
Total tax revenue and marginal tax revenue
Exercise
• Cigarette tax yields revenue R(t) = 50 + 25t – 75t2
• What is marginal revenue?
• What tax rate maximizes revenues?
• Why is this a maximum?
25 75(2 ) 25 150dR
MR t tdt
*
25 150 0
25 /150 1/ 6
MR t
t
concave
Functions not everywhere differentiable
Differentiable Continuous
Demand and cost functions
Average vs. marginal
Difference quotient of a polynomial (1)
0x
( )1
x
x
2( )
2x
x xx
3
2 2( )3 3 ( )
xx x x x
x
4
3 2 2 3( )4 6 4 ( ) ( )
xx x x x x x
x
Exercise: y = 4x2 – 8x + 3
• Find roots (x such that y = 0).
• Derivative
• Extreme value
22 8 8 4(4)(3)4
2 2(4)
2 4 31 .5
2
b b acy
a
8 8dy
xdx
*8 8 0 1dy
x xdx
Exercise: y = 4x2 – 8x + 3
0.5 1.0 1.5 2.0
1
1
2
3
Exercise: y = 4x2 – 8x + 3
0.5 1.0 1.5 2.0
5
5
y
y’ = 8x - 8
Differential vs. derivative
• Derivative is rate of change as x 0
• Differential is change in y along tangent line
0lim '( )x
y y dyf x
x x dx
'( )dy f x dx
Differential approximation and actual change
0'( )dy f x dx
Exercise: y = 16 – 4x + x3
• What is rate of change?
• What is derivative?
• What is differential?
• Let x0 = 2; x = 8
• Let x0 = 2; x = .2
24 3dy
xdx
2(3 4)dy x dx
2
2 2
(3(2 ) 4)(8) 64
(3(2 ) 4 3(2)(8) 8 )(8) 960
y
y
2
2 2
(3(2 ) 4)(.2) 1.6
(3(2 ) 4 3(2)(.2) .2 )(.2) 1.848
y
y
2 24 3 3y
x x x xx
Exercise
investment (I) and cost of borrowing (r) 2( ) 600 150 400I f r r r
Compute: ( )I f r r
0 1 22%; .5%; 1%r r r
1596.5 597.16 .66I
2 0 2(800 150) (800(.02) 150)(.01) 1.34I r r
2
0600 150(.02) 400(.02 ) 597.16I
2
1600 150(.025) 400(.025 ) 596.5I
2
2600 150(.03) 400(.03 ) 595.86I
2595.86 597.16 1.3I
1 0 1(800 150) (800(.02) 150)(.005) .67I r r
