By Goursat E.

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15. The function z = + rx 2 and <j> Show y where fa + 2 sxy whatever be the functions 16. + x<j>(y/x) y~ n ^(y/x) ty + px + 2 satisfies the qy = equation n 2 z, \f/. that the function - x - ai 0! (x) | + x - a z fa (x) \ + + x - | an \ n (x), (x), (x), together with their derivatives, 0i (x), (x), 0n(x), are continuous functions of x, has a derivative which is discontinuous for x = a\ Oz an fa , , , n (x), , , 17. Find a relation between the first and second derivatives of the function =/(&!

Is From [Note. the relation l-P2 = Q2(l-x) (a) it follows that - 2 PP = Q [2 Q (l - x*) - (b) The equation by Q-] and where R\ Show where let (t) is x shows that A; is prime to P and ; (b) shows that P divisible all dif a polynomial of the fourth degree and P / Q is a rational function. ] Each root of the equation R(U/ V) = 0, since must cause VU and hence also dx/dt, UV Show is V a constant. is to vanish, Q be a polynomial of the fourth degree whose roots are = U / be a rational function of t, such that that the function [Note.

3 v , fa fxz In general, a function of ; and so , \ on. p independent variables has just as many n as there are distinct terms in a homo of n in p independent variables that is, order geneous polynomial distinct derivatives of order ; as is shown in the theory of combinations. A Practical rules. certain number of practical rules for the cal culation of derivatives are usually derived in elementary books on * The notation /a Pyq (x, y, z) is used instead of the notation fxfyn z r (x, y, z) for Thus the notation fxy (x, y), used in place of fx y (x, y), is simpler and TRANS.

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A Course in Mathematical Analysis by Goursat E.


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