Differential Calculus II II. Higher Order Derivatives A. Introduction The second-order derivative, written f "(x) , measures the slope and the rate of change of the first derivative, just as the first derivative measures the slope and the rate of change of the original or primitive function. The third-order derivative f '"(x) measures the slope and rate of change of the second-order derivative, etc. Higher-order derivatives are found by applying the rules of differentia- tion to lower-order derivatives. Common notation: 2 d y nd 2 2 order derivatives: f "(x) y" D y 2 dx 3 d y rd 3 3 order derivatives: f "'(x) y"' D y 3 dx 4 d y th XXXXXXXXXX 4 order derivatives: f (x) y D y 4 dx Examples: 4 3 2 1) Find the successive derivatives of the following function: f (x) = 2x + 5x + 3x 3 2 f '(x) = 8x +15x + 6x 2 f "(x) = 24x + 30x + 6 f "'(x) = 48x + 30 (4) f (x) = 48 (5) f (x) = 0 For each of the following functions: a) find the second-order derivative; and b) evaluate it at x = 2. 3 2 2) y = 7x + 5x +12 dy 2 a) = 21x +10x dx 2 d y = 42x +10 2 dx b) At x = 2, 2 d y = XXXXXXXXXX = 94 2 dx 15x 3) y = 1- 3x dy (1- 3x)(5-) (5x - )( 3) a) = 2 dx (1- 3x) (5 - 15x-) - ( 15x) 5 = = 2 2 (1- 3x) (1 - 3x) 2 2 (1- 3x XXXXXXXXXXx)( 3) d y [ ] 30 - 90x XXXXXXXXXX3x) 30 = = = = XXXXXXXXXX dx (1- 3x) (1 - 3x) - (1 3x) - (1 3x) 2 d y XXXXXXXXXX b) At x = 2, = = = = - 2 3 3 dx XXXXXXXXXX125 25 Economic Application: If AR = f(Q), determine TR, MR and MR’. dTR TR = Q · AR(Q) = Q · f (Q); MR = = Q · f '(Q) + f (Q) dQ 2 d TR MR ' = = Q · f "(Q) + f '(Q) + f '(Q) = Q · f "(Q) + 2 f '(Q) 2 dQ Economic Interpretation in the case of a linear demand curve (not required for question 12): 1) AR(Q) = f (Q) is the demand curve 2) f '(Q) is the slope of the demand curve 3) Since f '(Q) <><>
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