Differentiation Rules Can these rules be extended to $\Bbb R^n$ with $n>1$ using norms? Or does that not work? From Calculus 1, we know these rules are applicable because we are working with 1 variable functions but how about when we have higher dimensions?
 A: The classic generalization of derivatives most intuitively similar to higher dimensions is partial derivatives.
The notion of a derivative is inherently (up through real analysis) tied to the idea of a limit. Recall
$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
defines the derivative of $f$ at $x$.
If we extend to $\mathbb R^n$ ($n>1$) and set $x, h \in \mathbb R^n$ for $f: \mathbb R^n \to \mathbb R^n$, This limit is not defined as we don't have a rigorous notion of dividing two length-$n$ vectors.
Your question about using norms to define derivatives doesn't specify a specific construct, so let's look at a few.
In general, we want the derivative operator to return real numbers (both positive and negative) and ensure that the 1-dimensional case can "fall out" under simplification.
$$ f'(x) = \lim_{h \to 0} \frac{||f(x+h) - f(x)||}{||h||}$$
We run into a first immediate problem with signs - all the 'derivatives' here must be positive (as both norms are always positive), so we could never have a negative derivative.
If we want Calculus 1 rules to fall out, this type of construct simply won't work. You can try this out with your favorite polynomial using $|| \cdot || = |\cdot|$ as a norm on $\mathbb R$.
The closest generalization of a derivative that uses norms is the total derivative of $f: \mathbb R^n \to \mathbb R^m$, defined implicitly as $f'(x)$ such that
$$ \lim_{\mathbf h\to \mathbf 0} \frac{||f(x+h) - (f(x) + f'(x) \mathbf{h}) ||}{|| \mathbf{h} ||} = 0$$
where $h \in \mathbb R^n$ is a vector. As $f(x)$ is a vector, $f'(x)$ is a $m \times n$ matrix. If $n=m=1$, then we have a derivative in in the classical sense using $| \cdot |$ as a norm. How do we compute $f'(x)$ then? It's actually a jacobian matrix, so we can build this concept using partial derivatives.
