16. Suppose f : R
n → R
n
is differentiable and has a differentiable inverse f
−1
: R
n → R
n
. Show that
D


The problem is about showing a relationship involving the derivative (or the Jacobian matrix) of a differentiable function \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) and its inverse \( f^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n \).

### Problem Statement:
Given a differentiable function \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) with a differentiable inverse \( f^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n \), show that:

\[
D(f^{-1})(f(x)) = (Df(x))^{-1},
\]

where \( Df(x) \) represents the Jacobian matrix of \( f \) at \( x \), and \( D(f^{-1})(f(x)) \) represents the Jacobian matrix of the inverse function \( f^{-1} \) at \( f(x) \).

### Proof:

#### Step 1: Consider the Chain Rule
Let \( y = f(x) \). Then by the definition of the inverse function, \( f^{-1}(y) = x \).

Consider the composition \( f \circ f^{-1} \). Since \( f \) and \( f^{-1} \) are inverses, we have:

\[
(f \circ f^{-1})(y) = y.
\]

Now, take the derivative of both sides with respect to \( y \). Using the chain rule, we obtain:

\[
D(f \circ f^{-1})(y) = Df(f^{-1}(y)) \cdot D(f^{-1})(y).
\]

But \( f \circ f^{-1} \) is the identity function \( \text{id}(y) = y \), whose derivative is the identity matrix \( I_n \). Thus:

\[
Df(f^{-1}(y)) \cdot D(f^{-1})(y) = I_n.
\]

This equation implies that the Jacobian matrix \( Df(f^{-1}(y)) \) is the inverse of the Jacobian matrix \( D(f^{-1})(y) \):

\[
Df(f^{-1}(y)) \cdot D(f^{-1})(y) = I_n \quad \text{or equivalently,} \quad D(f^{-1})(y) = (Df(f^{-1}(y)))^{-1}.
\]

#### Step 2: Substitution to Express in Terms of \( x \)
Since \( y = f(x) \), substitute \( y = f(x) \) into the previous equation:

\[
D(f^{-1})(f(x)) = (Df(x))^{-1}.
\]

This completes the proof.

### Conclusion:
We have shown that the Jacobian matrix of the inverse function \( f^{-1} \) at \( f(x) \) is the inverse of the Jacobian matrix of \( f \) at \( x \). Formally, this is expressed as:

\[
D(f^{-1})(f(x)) = (Df(x))^{-1}.
\]

Would you like more details or have any specific questions? Here are some follow-up questions:

1. How does the chain rule apply when dealing with compositions of functions and their inverses?
2. What are the conditions under which the inverse of a function is differentiable?
3. Can this result be extended to non-square Jacobian matrices, where the function maps between spaces of different dimensions?
4. How does the Inverse Function Theorem relate to this result?
5. What are some practical applications of this result in multivariable calculus and differential geometry?

**Tip:** Understanding the relationship between a function's Jacobian and that of its inverse is crucial in multivariable calculus, especially when dealing with coordinate transformations.

Explore the proof showing the relationship between the Jacobian matrix of a differentiable function and its inverse. Understand how the inverse Jacobian matrix relates to the derivative of the function in multivariable calculus. Suitable for advanced undergraduate students and researchers.

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