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### Chain Rule Notation

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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?

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
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.

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

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/
```
Associated Topics:
High School Calculus

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