Chain Rule Notation
Date: 04/21/99 at 04:55:25 From: Eric Subject: Chain Rule I'm trying to figure out these questions: Formula : (f*g)'(x)= g'(x) . f'[g(x)] 1) f(x) = 2x+6 g(x) = 3x-4 (f*g)'(x) = 3 . 2 = 6 I know that g'(x)= 3 but how about f'[g(x)]? How does 2 come about? I don't understand how it's done. 2) g(x) = 2x^2 + 5 h(x) = x^4 (g*h)'(x) = 4x^3 . 4x^4 = 16x^7 It's the same here. I know how to differentiate h(x) but I got stuck on g'[h(x)]. How does 4x^4 come by? Please help me, Thanks.
Date: 04/21/99 at 10:19:17 From: Doctor Mitteldorf Subject: Re: Chain Rule Dear Eric, The chain rule can be taught in such a way that it's quite transparent, or it can be made utterly mysterious with bad notation. It looks as if you've been a victim of the latter. The chain rule is about taking the derivative of a function of a function. Instead of f being a function of x, we have f is a function of g, and g is a function of x. In this notation, the chain rule can be written: df/dx = df/dg . dg/dx It seems almost obvious when you write it that way. Just "cancel out" the dg's in the numerator and denominator. In your example (1), f(x) = 2x+6 g(x) = 3x-4 The teacher gave you a notation that's deliberately confusing. You have to remember that the x in these equations is a dummy variable. The top equation just says f is a function that takes its argument, multiplies it by 2, then adds 6. The x is there just as a placeholder. You can replace it with a or b or theta or phi and the equation says exactly the same thing. But in this case, you want to replace it with g: f(g) = 2g+6 Is it obvious why I want to replace the x by a g? It's because f*g means "f composed with g," or "the function f taken of the function g." (Using the * for "composed with" can be a confusing notation too if you are mixing it in an equation in which * means simply multiplication. You have wisely used a dot for multiplication.) Coming back now to problem 1, let's do it two ways. First, we'll actually find f*g and differentiate it. Second, we'll use the chain rule. Then we'll be in a position to check that the two answers are the same. First, f(g) = 2g+6 g(x) = 3x-4 Substituting the second equation into the first, you have f(x) = 2(3x-4)+6 = 6x-2 It's obvious, then, that f'(x) = 6. Second, we'll use the chain rule: df/dx = df/dg . dg/dx = 2 . 3 = 6 Hence, we get the same answer both ways. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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