Advanced calculus; . TM = subtangent = yjy\, MNOT = ^--intercept of tangent == x The derivation of these results is sufficiently evi-dent from the figure. It may be noted that thesubtangent, subnormal, etc., are numerical valuesfor a given point of the curve but may be regardedas functions of x like the geometrical and physical problems it is frequently necessary toapply the definition of the derivative to finding the derivative of anunknown function. For instance if A denote thearea under a curve and measured from a fixedordinate to a variable ordinate, A is surely a func-tion A
Advanced calculus; . TM = subtangent = yjy\, MNOT = ^--intercept of tangent == x The derivation of these results is sufficiently evi-dent from the figure. It may be noted that thesubtangent, subnormal, etc., are numerical valuesfor a given point of the curve but may be regardedas functions of x like the geometrical and physical problems it is frequently necessary toapply the definition of the derivative to finding the derivative of anunknown function. For instance if A denote thearea under a curve and measured from a fixedordinate to a variable ordinate, A is surely a func-tion A (x) of the abscissa x of the variable the curve is rising, as in the figure, then MPQM < AA< MQPM, or yAx < AA <(jj + Ay) by Ax and take the limit when Ax = 0. There results. lim y Ax = 0 Hence lim —— g Aa: = 0 Ax lim — = Ax = 0 Ax lim (y + Ay). Ax = 0 dAdx (43) Rollers Theorem and the Theorem of the Mean are two importanttheorems on derivatives which will be treated in the next chapter butmay here be stated as evident from their geometric Theorem states that: If a function has a derivative at every
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