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Theorem 15.3 (Chain Rule): Assume that f is differentiable at x and that g is differentiable at f (x). Then the composition g f is differentiable at x, and its derivative (g f ) is given by (g f ) (x) = g ( f (x))f (x) that is, Dx (g( f (x))) = g (f (x))Dx f (x) The proof of Theorem 15.3 is tricky; see Problem 15.27. The power chain rule (Theorem 15.2) follows from the chain rule (Theorem 15.3) when g(x) = x n . Applications of the general chain rule will be deferred until later chapters. Before leaving it, however, we shall point out a suggestive notation. If one writes y = g( f (x)) and u = f (x), then y = g(u), and (15.2) may be expressed in the form dy du dy = dx du dx (15.3) (15.2)
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just as though derivatives were fractions (which they are not) and as though the chain rule were an identity obtained by the cancellation of the du s on the right-hand side. While this identity makes for an easy way to remember the chain rule, it must be borne in mind that y on the left-hand side of (15.3) stands for a certain function of x [namely (g f )(x)], whereas on the right-hand side it stands for a different function of u [namely, g(u)]. EXAMPLE Using (15.3) to rework the preceding example (c), we write
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y = 3x 5 where u = 3x 5. Then, dy du 12 12 dy = = ( 4u 5 )(3) = 5 = 5 dx du dx u 3x 5
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Differentiation of Rational Powers We want to be able to differentiate the function f (x) = x r , where r is a rational number. The special case of r an integer is already covered by Rule 7 of 13.
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algebra A rational number r is one that can be represented in the form r = n/k, where n and k are integers, with k positive. By de nition, an/k = ( k a)n except when a is negative and k is even (in which case the kth root of a is unde ned). For instance, 3 (8)2/3 = ( 8)2 = (2)2 = 4 1 1 5 (32) 2/5 = ( 32) 2 = (2) 2 = 2 = 4 2 3 ( 27)4/3 = ( 27)4 = ( 3)4 = 81 ( 4)7/8 is not de ned Observe that k ( k a)n = an
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THE CHAIN RULE
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whenever both sides are de ned. In fact, (( k a)n )k = ( k a)nk = ( k a)kn = (( k a)k )n = an which shows that ( k a)n is the kth root of an . In calculations we are free to choose whichever expression for an/k is the more convenient. Thus: (i) 642/3 is easier to compute as 3 ( 64)2 = (4)2 = 16 than as but (ii) ( 8)2/3 is easier to compute as
(64)2 =
than as
3 ( 8)2 = 8 = 2 3 2 8
The usual laws of exponents hold for rational exponents: (1) ar as = ar+s (2) (3) (4) ar = ar s as ar ab
= ars = a r br
where r and s are any rational numbers.
Theorem 15.4: For any rational number r, Dx (x r ) = rx r 1 . For a proof, see Problem 15.6. EXAMPLES
1 1 1 1 (a) Dx ( x) = Dx(x 1/2 ) = x 1/2 = = 2 2 x 1/2 2 x (b) Dx (x 3/2 ) = 3 1/2 3 x = x 2 2 (c) Dx (x 3/4 ) = 3 1/4 3 1 3 x = = 4 4 x 1/4 44x
Theorem 15.4, together with the chain rule (Theorem 15.3), allows us to extend the power chain rule (Theorem 15.2) to rational exponents. Corollary 15.5: If f is differentiable and r is a rational number, Dx (( f (x))r ) = r( f (x))r 1 Dx f (x) EXAMPLES
(a) Dx x 2 3x + 1 = Dx ((x 2 3x + 1)1/2 ) = = 1 2 (x 3x + 1) 1/2 Dx (x 2 3x + 1) 2
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