2. We first note
Thus we find
But specifically, we are interested in
Plugging this in allows us to write
Observe under our substitution, we have the and other big O terms be gathered into the term.
So what? Observe
This is precisely the chain rule. Similarly, we find
3. Implicit Differentiation Revisited. Recall implicit differentiation required us to find from some complicated expression like
What to do? First we write
Next we say . So we find
where we set the derivative of to be zero since it’s equal to the derivative of zero. We can then write (taking )
and divide both sides by to get
But this is precisely what implicit differentiation gives us!
4. Warning for Physicists. Physicists often use partial derivative notation slightly differently. If is the position of a particle, and is its momentum, physicists consider arbitrary functions of the form
This is strictly speaking not quite true. The error committed lies in treating and as functions of time: really they are variables whom we are trying to express as functions of time.
Exercise 1. Let where , , and . Find .
Exercise 2. Let where and . Find and .
Exercise 3 [Math.SE]. Consider the equation . What is ?