Leibniz's differential notation

leads 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

Then, we have

which is the Chain Rule.

watko@mit.edu Last modified August 26, 1998