We saw also in ex xxxvii 7 that the types of

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We saw also in Ex. xxxvii . 7 that the types of discontinuity which occur most commonly, when we are dealing with the very simplest and most ob- vious kinds of functions, such as polynomials or rational or trigonometrical functions, are associated with a relation of the type φ ( x ) +
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[VI : 112] DERIVATIVES AND INTEGRALS 243 O X Y P Q Q R R ( a ) P Q Q R R ( b ) P Q Q R R ( c ) P Q Q R R ( d ) Fig. 38. or φ ( x ) → -∞ . In all these cases, as in such cases as those considered above, there is no derivative for certain special values of x . In fact, as was pointed out in § 111 , (1), all discontinuities of φ ( x ) are also discontinuities of φ 0 ( x ). But the converse is not true, as we may easily see if we return to the geometrical point of view of § 110 and consider the special case, hitherto left aside, in which the graph of φ ( x ) has a tangent parallel to OY . This case may be subdivided into a number of cases, of which the most typical are shown in Fig. 38 . In cases ( c ) and ( d ) the function is two valued on one side of P and not defined on the other. In such cases we may consider the two sets of values of φ ( x ), which occur on one side of P or the other, as defining distinct functions φ 1 ( x ) and φ 2 ( x ), the upper part of the curve corresponding to φ 1 ( x ). The reader will easily convince himself that in ( a ) { φ ( x + h ) - φ ( x ) } /h + , as h 0, and in ( b ) { φ ( x + h ) - φ ( x ) } /h → -∞ ; while in ( c ) { φ 1 ( x + h ) - φ 1 ( x ) } /h + , { φ 2 ( x + h ) - φ 2 ( x ) } /h → -∞ ,
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[VI : 113] DERIVATIVES AND INTEGRALS 244 and in ( d ) { φ 1 ( x + h ) - φ 1 ( x ) } /h → -∞ , { φ 2 ( x + h ) - φ 2 ( x ) } /h + , though of course in ( c ) only positive and in ( d ) only negative values of h can be considered, a fact which by itself would preclude the existence of a derivative. We can obtain examples of these four cases by considering the functions defined by the equations ( a ) y 3 = x, ( b ) y 3 = - x, ( c ) y 2 = x, ( d ) y 2 = - x, the special value of x under consideration being x = 0. 113. Some general rules for differentiation. Throughout the theorems which follow we assume that the functions f ( x ) and F ( x ) have derivatives f 0 ( x ) and F 0 ( x ) for the values of x considered. (1) If φ ( x ) = f ( x ) + F ( x ) , then φ ( x ) has a derivative φ 0 ( x ) = f 0 ( x ) + F 0 ( x ) . (2) If φ ( x ) = kf ( x ) , where k is a constant, then φ ( x ) has a derivative φ 0 ( x ) = kf 0 ( x ) . We leave it as an exercise to the reader to deduce these results from the general theorems stated in Ex. xxxv . 1. (3) If φ ( x ) = f ( x ) F ( x ) , then φ ( x ) has a derivative φ 0 ( x ) = f ( x ) F 0 ( x ) + f 0 ( x ) F ( x ) . For φ 0 ( x ) = lim f ( x + h ) F ( x + h ) - f ( x ) F ( x ) h = lim f ( x + h ) F ( x + h ) - F ( x ) h + F ( x ) f ( x + h ) - f ( x ) h = f ( x ) F 0 ( x ) + F ( x ) f 0 ( x ) .
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[VI : 113] DERIVATIVES AND INTEGRALS 245 (4) If φ ( x ) = 1 f ( x ) , then φ ( x ) has a derivative φ 0 ( x ) = - f 0 ( x ) { f ( x ) } 2 . In this theorem we of course suppose that f ( x ) is not equal to zero for the particular value of x under consideration. Then φ 0 ( x ) = lim 1 h f ( x ) - f ( x + h ) f ( x + h ) f ( x ) = - f 0 ( x ) { f ( x ) } 2 . (5) If φ ( x ) = f ( x ) F ( x ) , then φ ( x ) has a derivative φ 0 ( x ) = f 0 ( x ) F ( x ) - f ( x ) F 0 ( x ) { F ( x ) } 2 .
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