TITLE: What is the Implicit Function Theorem good for?
QUESTION [51 upvotes]: What are some applications of the  Implicit Function Theorem  in calculus?  The only applications I can think of are:

the result that the solution space of a non-degenerate system of equations naturally has the structure of a smooth manifold;
the Inverse Function Theorem.

I am looking for applications that would be interesting to an advanced US math undergraduate 
or 1-st year graduate student who is seeing the Implicit Function Theorem for the first or second time.  The context is I will be explaining this result as part of 
a review of manifold theory.

REPLY [7 votes]: I was just sitting down to brainstorm some applications, since we're going to finish the proof next week and I want to show some payoff. Here is what I came up with, not counting ideas already mentioned in other answers (many of which I like):

The orthogonal group is a manifold, of dimension $\binom{n}{2}$. It's not too hard to compute the derivative of the map $X \mapsto X X^T$ from $n \times n$ matrices to $n \times n$ symmetric matrices and show that it is rank $\binom{n+1}{2}$ when $X$ is invertible. So the preimage of $\mathrm{Id}$ is a manifold of dimension $n^2 - \binom{n+1}{2}$.
Suppose I have a smooth map $(x,y) \mapsto (f,g)$ from $\mathbb{R}^2 \to \mathbb{R}^2$, so that $\mathrm{rank} (Df)$ is everywhere $1$. Then one of $f$ and $g$ is locally a function of the other. (Proof: After some initial fiddling around, we can assume that $\partial f/\partial x \neq 0$. Then $f=\mathrm{constant}$ is a curve. Differentiating $g$ along this curve gives $0$.) I remember this trick being used to great effect in Kenyon-Okounkov to show that two functions obeyed a relation without actually constructing the relation; I'll try to find some simple examples where it is very nonobvious what the relation is.
This top voted proof of the fundamental theorem of algebra uses centrally that a submersion is an open map, which is the key lemma in all proofs of IFT that I know. 

I'll come back in a few weeks and report on how these went.