Problem 2-13 from Spivak's Calculus on Manifolds asks the following:
Problem 2-13. Define $IP : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ by $IP(x,y) = \langle x, y \rangle$.
- Find $D(IP)(a,b)$ and $(IP)'(a,b)$.
- If $f,g : \mathbb{R} \to \mathbb{R}^n$ are differentiable and $h : \mathbb{R} \to \mathbb{R}$ is defined by $h(t) = \langle f(t), g(t) \rangle$, show that $$h'(a) = \langle f'(a)^T, g(a) \rangle + \langle f(a), g'(a)^T \rangle.$$
- If $f : \mathbb{R} \to \mathbb{R}$ is differentiable and $|f(t)| = 1$ for all $t$, show that $\langle f'(t)^T, f(t) \rangle = 0$.
- Exhibit a differentiable function $f : \mathbb{R} \to \mathbb{R}$ such that the function $|f|$ defined by $|f|(t) = |f(t)|$ is not differentiable.
What I find weird is that part (4) of the problem seems quite unrelated to the previous three parts of the question. The first three parts are specifically about the inner product as a multilinear function and the properties it and its derivative have, whereas for part (4) I can just take $f(x) = x$ and be done with it.
Typically, Spivak's problems are set up in a way to motivate some deeper idea. It is possible that he has slipped up here (after all, Calculus on Manifolds has numerous typos as well). But, I'd still like to ask,
Is there a way to look at part (4) of this problem in such a way that it connects to the inner product (or more precisely, to the previous three parts of the problem)?
Okay, here's the best that I am able to do, even though it is just a disguised way of saying that $x \mapsto \lvert x \rvert$ is not differentiable at $0$.
Choose the functions $f$ and $g$ from part (2) as $f(x) = (x,x,\dotsc,x)$ and $g(x) = (1,1,\dotsc,1)$ for all $x \in \mathbb{R}$.
Then $f$ and $g$ are differentiable, and so is $h(t) = \langle f(t),g(t) \rangle = nt$. But, $\lvert h \rvert (t) = n \lvert t \rvert$, which is not differentiable at $0$. Hence, $\lvert h \rvert$ is not differentiable even though $h$ is differentiable.