The Chain Rule
Read this section to learn about the Chain Rule. Work through practice problems 1-8.
Chain Rule (Leibniz notation form)
If is a differentiable function of , and is a differentiable function of , then is a differentiable function of and .
Although this nice short argument gets to the heart of why the Chain Rule works, it is not quite valid. If , then it is possible to show that for all very small values of , and the "idea for a proof" is a real proof. There are, however, functions for which for lots of small values of , and these create problems for the previous argument. A justification which is true for ALL cases is much more complicated.
The symbol is a single symbol ( as is ), and we cannot eliminate du from the product in the Chain Rule. It is, however, perfectly fine to use the idea of eliminating du to help you remember the statement of the Chain Rule.
Solution: so so . Finally, using the Chain Rule,
There is also a composition of functions form of the Chain Rule. The notation is different, but it means precisely the same as the Leibniz form.