Schaum's Outline of Theory and Problems of Beginning Calculus

Chapter 15The Chain RulelS.lCOMPOSITE FUNCTIONSThere are still many functions whose derivatives we do not know how to calculate; for example,(i) J- (ii) $Z7 (iii) (x2 + 3x - 1)23In case (iii), we could, of course, multiply x2 + 3x - 1 by itself 22 times and then differentiate theresulting polynomial. But without a computer, this would be extremely arduous.The above three functions have the common feature that they are combinations of simpler functions:(i) d m ’is the result of starting with the functionf(x) = x3 - x + 2 and then applying thefunction g(x) = fi to the result. Thus,Jx’ - x + 2 = g(f(x))(ii) 15 is the result of starting with the function F(x) = x + 4 and then applying the functionG(x) = fi. Thus,dZ4 = G(F(x))(iii) (x2 + 3x - 1)23 is the result of beginning with the function H(x) = x2 + 3x - 1 and then applyingthe function K(x) = x23. Thus,(x2 + 3x - 1)23 = K(H(~))Functions that are put together this way out of simpler functions are called composite functions.Definition: Iff and g are any functions, then the composition g 0 f off and g is the function such thatThe “process” of composition is diagrammed in Fig. 15- 1.(9 O f K 4 = g(f(x))Fig. 15-1EXAMPLES(a) Letf(x) = x - 1 and g(x) = x2. Then,(9 0 fXx) = g(f(x)) = g(x - 1) = (X -On the other hand,(f O sKx) = f(g(4) =f(x2) = x2 - 1Thus,fo g and g ofare not necessarily the same function (and usually they are not the same).116