**1. Problem.** Consider a function where we parametrize

(1)

If , how does change? We will need to use partial derivatives and Taylor expansion…

**2.** We first note

(2)

Similarly

(3)

Thus we find

(4)

But specifically, we are interested in

(5)

and

(6)

Plugging this in allows us to write

(7)

as

(8)

Observe under our substitution, we have the and other big O terms be gathered into the term.

So what? Observe

(9)

This is precisely the chain rule. Similarly, we find

(10)

**3. Implicit Differentiation Revisited.** Recall implicit differentiation required us to find from some complicated expression like

(11)

What to do? First we write

(12)

Next we say . So we find

(13)

where we set the derivative of to be zero since it’s equal to the derivative of zero. We can then write (taking )

(14)

and divide both sides by to get

(15)

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

(16)

and write

(17)

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 ?