The Chain Rule - a more formal approach
Suggested Prerequesites:
The definition of the derivative,
The chain rule
Leibniz's differential notiation
lead us
to consider treating derivatives as fractions, so that given a composite
function y(u(x)), we guess that
This speculation turns out to be correct, but we would like a better
justification that what is perhaps a happenstance of notation. Let's
start with the definition of the derivative and try to arrive at this
result:
Given: y = f(u(x))
By simple algebra, we know that
Then:
Differentiablility implies continuity, therefore
as
.
And we're home free.
A less formal look at the chain rule (including
examples and exercises)
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jjnichol@mit.edu
Last modified 23 June 1997