Hãy xem xét hình , và sử dụng định nghĩa nói trên để tính giá trị của độ dốc của AB dây. Như điểm B mang lại tùy tiện gần gũi hơn với điểm A, tuy nhiên, giá trị của độ dốc của AB phương pháp tiếp cận giá trị của độ dốc tại điểm duy nhất, | 62 INTRODUCTION To MATHEMATICAL ECONOMICS DERIVATIVE OF A FUNCTION Consider again the function v x The slope ofthisfunctionis nehnedasthe change in the value of y divided by a change in the value of X or the rise over the -run When defining the slope between two discrete points the formula for the slope may be given as Slope s f v. u Consider Figure and use the foregomg dehnition to calculate the value of the slope of the cord AB. As point B is brought arbitrarily closer to point A however the value of the slope of AB approaches the value of the slope afthesmgle poiet 1 wtiichwuuldbccu uivalcnlt theslopeof a tangent tvtOe cuuvealthalpoiog. This procedhrs isgreotfysimpliheg howeveribyhtil in dnrivalive of dui toaelSen sndcalseiatlngies value in ttũscaeeiatxi. The htslderivatis e oUafunettontmylyoiis simolytheflope ol thefunc-tion when tSemlerval alonethe horiz nelataxts e tween hi andxetis made inhnitesimetly amail. Technicallly the derivative is the limit of the ratio Ay Av as Ax approacheszef e i e h iitnl - 1 da aT AxJ When the limit of a function as X - x0oquals the Value of the function at Xo the function is said to be continuous ot x0 that is limx xo fix f x0 . FIGURE Discrete versus instantaneous rates of change. RULES OF DIFFERENTIATION 63 Calculus offers a set of rules for using derivatives slopes for making optimizing decisionssuch asminimizing cost TC or maximizing total profit rt . RULES OF DIFFERENTIATION Having established that the derivative of a function is the limit of the ratio of the change in the dependentviriabletothechangein the independent variable we will now enumerate some general rules of differentiation that willbt of considerablcvaluctlirougliout tdcrnmiOndo r dfthis course. ItshouldbeuodnrsfoneCthrtforrfnnctionnobedidferonniablh at a point ifmrbnCrcwdUbefined thnlis ttmuftdecontinabUfor rmoodh. It is not poenibleCofìndthederioatt nofcfunctionradlisdidbObtinuuuf . has o obrneo
