Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis", 3rd ed., Prentice Hall. Fundamental Methods of Mathematical Economics, Kevin Wainwright, Alpha Chiang. Mathematics for Economists, Carl P. Simon, Lawrence E. Blume

Derivatives - introduction Suppose a car is accelerating from 30mph to 50mph. At some point it hits the speed of 40mph, but when? Speed = (distance travelled)/(time passed) How is it possible to define the speed at a single point of time?

Approximations Idea: find distance/time for smaller and smaller time intervals. Here f shows how the distance is changing with the time, x. The time difference is h while the distance travelled is f(x+h) – f(x).

The derivative Our approximation of the speed is As h gets smaller, the approximation of the speed gets better. When h is infinitesimally small, the calculation of the speed is exact. Leibniz used the notation dy/dx for the exact speed.

Example f(x) = x 2 f(x+h) – f(x) = (x + h) 2 – x 2 = (x 2 + 2xh + h 2 ) - x 2 = 2xh + h 2 Speed = (2xh + h 2 )/h = 2x + h When h is infinitesimally small, this is just 2x.

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero. Rules for Differentiation

We saw that if,. This is part of a pattern. examples: power rule Rules for Differentiation

Proof:

examples: constant multiple rule: Rules for Differentiation

(Each term is treated separately) constant multiple rule: sum and difference rules: Rules for Differentiation

product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as: Rules for Differentiation

quotient rule: or Rules for Differentiation

Formulas you should learn ( Cx a )’=Cax (a-1) ; C, a – a real number (e x )’=e x (a x )’=a x lna; a>0

Derivatives rules - summary,,,, for c is a constant,

Consider a simple composite function: Chain Rule

Chain Rule: example: Find: Chain Rule If is the composite of and, then:

Differentiation of Multivariate Functions The partial derivative of a multivariate function f(x,y) with respect to x is defined as

Differentiation of Multivariate Functions f(x 1,x 2 )= Cx 1 a x 2 b