advertisement

Multiplicative constant: if y = kf (x) then dy = kf 0 (x) dx Derivatives in Economics This leaflet is an overview of differentiation and its applications in Economics. The marginal cost M C is the rate of change of the total dy 0 0 = f (x) ± g (x) cost function T C: M C = dT C , where q is the output. Addition rule: if y = f (x) ± g(x) then dx dq Similarly, the marginal revenue M R is the rate of change Product rule: if y = f (x) × g(x) then dT R of the total revenue function T R: M R = . When dy 0 0 dq = f (x)g(x) + f (x)g (x) M R is positive, T R is an increasing function of q, and dx when M R is negative, T R is a decreasing function of q. f (x) Quotient rule: if y = then The elasticity E of a function q = f (p) is the rate of prog(x) portionate change in q given a proportionate change in p: dq 0 0 dy f (x)g(x) − f (x)g (x) d ln q q = E = dp = . This is the slope of the function when dx (g(x))2 d ln p p plotted on a log-log scale. Chain rule (derivative of a function of a function): Author: Morgiane Richard, University of Aberdeen if y = g(u) with u = f (x) then community project mathcentre community project encouraging academics to share maths support resources All mccp resources are released under a Creative Commons licence mcccp-richard-5 For the help you need to support your course Differentiation for Economics and Business Studies Functions of one variable Reviewer: Anthony Cronin, University College Dublin The derivative of a function f is a new function obtained df by differentiating f . It can be written f 0 or . It is the dx rate of change of f and gives information on the shape and optimum values of f . Table of Derivatives y = f (x) k constant x x2 xn ex ln x eax+b ln (ax + b) ln (f (x)) dy = f 0 (x) dx 0 1 2x nxn−1 ex 1 x aeax+b a ax0 + b f (x) f (x) Rules of Differentiation For any function f and g and any constant value k: Additive constant: if y = f (x) + k then dy = f 0 (x) dx 2 dy dy du = = g 0 (u) × f 0 (x) dx du dx Shape of Function sign of dy = f 0 (x) dx >0 >0 <0 <0 sign of d2 y = f 00 (x) dx2 >0 <0 >0 <0 1.5 concave decreasing function 1 shape of the curve of f 0.5 convex decreasing function increasing and convex increasing and concave decreasing and convex decreasing and concave Stationary points 0 0 0.5 1 1.5 2 Figure 1: Examples of decreasing concave and convex functions 4 First Order Condition (FOC): if a point x0 is such that f 0 (x0 ) = 0, then it is a stationary point. It can be a maximum, or a minimum, or an inflection point. Second Order Condition (SOC): the sign of the second derivative indicates whether the optimum is a maximum, minimum or inflection point: 3.5 convex increasing function 3 2.5 2 1.5 concave increasing function 1 0.5 value of dy (x0 ) = f 0 (x0 ) dx 0 0 0 sign of d2 y (x0 ) = f 00 (x0 ) dx2 >0 <0 0 Nature of point at x0 minimum maximum inflection 0 0 1 2 3 5 Figure 2: Examples of increasing concave and convex functions www.mathcentre.ac.uk 1 4