Mathematics

# Derivatives & Differentiation – Introduction

“Government regulations generally limit the number of fish taken from a given fishing ground by commercial fishing boats in season.” (Haeussler, Paul, & Wood, 2008) This assumes market failure and that governments can predict fish depletion more accurately than fisheries can, in order to ensure there are adequate levels of fish to be sustainable in maximizing the # of fish population available to harvest. “For a given fish habitat, [they use a mathematical function called the reproductive curve estimating] a fish population a year from now, P(n+1), based in the population now, P(n), assuming no external intervention (i.e., no fishing, no influx of predators, and so on).” (Haeussler et al., 2008) If I were to graph this it would look like: This short article is an introduction on derivatives and differentiation and their economic applications. The importance of relating these concepts to real economic examples is an easy way to help understand the mathematics. Differentiation is basically a method used to find the derivative of a function at a given point. Represented as f(x) the derivative of f with respect to the x variable, measuring the point of intersection to find the relative extrema, rate of change and slope of the original function. To begin we will cover the derivative of a function and in the next article we will cover the different rules to find derivatives and apply each to a set of examples. A problem in finding the slope of the tangent line, to find a derivative of a function two conditions must be met: (1) the continuity, and (2) the smoothness. Two conditions (called the rules of optimization) in order to find minimum profit and maximum profit, and minimum cost, are:

First order condition– A test to find one or more maximum or minimum profit points of a function. A condition to find maximum profit is that the marginal value or slope of the curve at the point must be equal to 0 with no change in y. Another condition is that the function y=f(x) means the derivative dY/dX point is equal to zero. f'(x)=0, set to 0 to solve for x. Where the critical value of (a) of x shows where f(x) reaches a minimum or maximum. Graphing the increase and decrease in the function: Second order condition–  A test to find out whether the point determined by the FOC is the maximum or a minimum point. By just setting the derivative to 0 and solving for the equation does not determine whether the point is a maximum or minimum point. To test we need to find the derivative of the derivative, that is the 2nd derivative and see if a is < or > than 0. f”(x)>0 => reactive minimum turning point, and f”(x)<0 => reactive maximum turning point. Graphing the relative extrema, to show if the slope is < or > than 0. Here’s, a quadratic representation of the cost function: Example (i)-  y=2x^3-30x^2+126x+59

Find maximum and minimum:

First order of condition (FOC): find the first derivative,      y1=6x^2-60x+126=0

Set to zero, 6x^2-60x+126=0

=>  6(x^2-10x+21)=0  =>  x^2-10x+21=0  => -7x and -3x =12×2

1st and 3rd terms, x^2-7x-3x+21=0

=>  x(x-7)-3(x-7)=0  =>  (x-7)(x-3)=0  =>  x-7=0, or, x-3=3

so, the critical values are x=7 and x=3

Second order of condition (SOC): find the second derivative to differentiate the function again to find which are the maximum and minimum points.       y2=12x-60

When, x=7  =>  y2=12(7)-60   =>  84-60   =>  24>0

*reaches minimum turning point when x reaches or is close to 7. y2=(7)>0

=>  fn is at a minimum when x=7.

When, x=3  =>  y2=12(3)-60 =>  36-60 =>  -24<0

=>  fn measures a relative maximum when x=3.

Reference List (American Psychological Association)

Haeussler, E., Paul, R., & Wood, R. (2008). Introductory mathematical analysis: for business economics, and the life and social sciences. (12th ed.). United States of America: Pearson Education Inc.

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