Posted by Brad Paul on October 28, 2002 at 15:56:15:
In Reply to: chain rule posted by Olivia on October 28, 2002 at 04:35:08:
: I just do not understand chain rule. Can someone break into VERY layman terms and explain it to me very clearly? I also don't understand why its done...
Imagine two functions. Let these functions be f(x) and g(x).
First what does it mean to have a function? A function is a black box
that when given a number (which we call x) returns just one
number. This one number is different for different values of x. Next we
may ask how does the returned number change when the value of x is
changed? Imagine giving one of our functions, say g(x) a value of x
that we slowly change and we watch how the output of g(x) changes. If
we notice that the output of g(x) changes fast when we slowing change
the input (x), we know that g(x) has a large derivative in this
vicinity. I can write the derivative of g(x) as g'(x).
Now because the output of g(x) is just a number let's take this number
and stick in into our function f. f is just a dumb function and does
not know this number came from the output of g(x) nor does it
care. We can write this in the following way: f(g(x)). Now if we want
to know how does f(g(x)) change when I change x we have to be
careful. First we need to know how f changes when it's input
changes. Remember f's input is g(x). This is just f'(g(x)). But this is
not what we want. We want how does the compound function f(g(x)) change
with x. To get this we need to multiply how f's input changes with
respect to it's input which is g'(x). Therefore the total change in
the compound function f(g(x)) is f'(g(x))g'(x).
Let's look at a simple example in words. I have a simple black box
with a big white letter g on it. When I give it a number it returns a
number that is twice as large. We know it's internal function is
g(x)=2 x we also know that g'(x)=2. Lets take the output of the the g
black box and put it into an other black box called f. The f black box
will always return a number that is three times larger that what it is
given. We know it's internal function is f(x)=3 x and we also know that
f'(x)=3. Now let's take the output of g and plug into the input of
f. We could say we are chaining the functions together. Now we ask how
does the output of f change when we change the input of g? We use the
chain rule. f'(g(x))g'(x)-> (3)(2)=6. We can check this because we are
clever and know the functions of the two boxes and can write
f(g(x))=3(2 x) = 6x which gives us a derivative of 6. This example was
very simple and in this case our compound function was something we
could find a derivative of right out.
The moral of the story: The chain rule always works but sometimes the
problem is so simple that it is not needed.
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