Why generalize the derivative for multivariable functions?

Question

Sorry if this is a dupe (did a search, couldn't find anything).

In single variable calculus, if the following limit exists:

$$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$

then this expression itself is the derivative of $f$ at $x$. This is nicely motivated geometrically and otherwise. This definition gives of $f'(x)$ which gives the best linear approximation $\tilde{f}$ to $f$ at $x$ by

$$\tilde{f}(t) = f'(x)t + f(x).$$

For functions of several variables, most of the time I see the derivative defined in terms of the best linear approximation to the function explicitly. Specifically, the derivative of a multivariable function $g$ at $\mathbf{y}$ is some linear operator $L(\mathbf{y})$. Concretely, if the following expression is satisfied:

$$\lim_{\mathbf{h} \rightarrow 0}\frac{\|g(\mathbf{y} + \mathbf{h}) - g(\mathbf{y}) + L(\mathbf{y})\|}{\|\mathbf{h}\|} = 0,$$

then $L(\mathbf{y})$ is the derivative of $g$ at $\mathbf{y}$. We can show that $L(\mathbf{y})$ is unique in this case.

My question is why is this generalization necessary? What is the problem with simply defining is analogously to the single variable case as

$$L(\mathbf{y}) = \lim_{\mathbf{h} \rightarrow 0}\frac{g(\mathbf{y} + \mathbf{h}) - g(\mathbf{y})}{\|\mathbf{h}\|}?$$

Answer 1

For one thing, that limit isn't typically going to exist, even when the function is differentiable. Take $$g(x,y) = x \, .$$

Consider $h^1_n = (\frac{1}{n},0)$, $h^2_n = (0,\frac{1}{n})$. If you take the limit along the sequence $(h^1_n)_{n=1}^\infty$, the limit is $1$. Taking it along $(-h^1_n)_{n=1}^\infty$, the limit is $-1$. Taking it along $(h^2_n)_{n=1}^\infty$, the limit is zero.

The candidate definition also doesn't agree with the standard definition of the derivative in the one-dimensional case, because of the norm (absolute value) in the denominator. You would have $g(x) = |x|$ being differentiable at zero, and $g(x) = x$ not differentiable.

Further, the standard definition generalizes for functions $g: \mathbb{R}^N \to \mathbb{R}^M $. Your proposal won't be very straightforward to generalize to the case where the range is non-scalar.