Could someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$?
$$\nabla\|x\|_2^2 = 2x$$
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$\begingroup$Use the definition. If $$f(x)=\|x\|^2_2= \left(\left(\sum_{k=1}^n x_k^2 \right)^{1/2}\right)^{2}=\sum_{k=1}^n x_k^2 ,$$ then $$\frac{\partial}{\partial x_j}f(x) =\frac{\partial}{\partial x_j}\sum_{k=1}^n x_k^2=\sum_{k=1}^n \underbrace{\frac{\partial}{\partial x_j}x_k^2}_{\substack{=0, \ \text{ if } j \neq k,\\=2x_j, \ \text{ else }}}= 2x_j.$$ It follows that $$\nabla f(x) = 2x.$$
$\endgroup$ 9 $\begingroup$Another approach that extends to more general settings is to use the connection between the norm and the inner product,$$\|x\|^2 = (x,x).$$
We have the finite difference,\begin{align} \|x+sh\|^2 - \|x\|^2 &= (x+sh,x+sh) - (x,x) \\ &= (x,x) + 2s(x,h) + s^2(h,h) - (x,x) \\ &= 2s(x,h) + s^2(h,h). \end{align}
The gradient acting in the direction $h$ is the limit of this finite difference as the stepsize goes to zero,\begin{align} (\nabla\|x\|^2, h) &:= \lim_{s \rightarrow 0} \frac{1}{s}\left[\|x+sh\|^2 - \|x\|^2\right] \\ &= \lim_{s \rightarrow 0} \frac{1}{s}\left[2s(x,h) + s^2(h,h)\right] \\ &= (2x,h). \end{align}Since this holds for any direction $h$, the gradient must be $\nabla \|x\|^2 = 2x$.
$\endgroup$ $\begingroup$I'm not sure if this is rigorous enough to count as a proof, but an elegant way to obtain derivatives of vector expressions is to use matrix differential calculus.
Let $y = \lVert x \rVert_2^2 = x^{T} x$ with $x \in \mathbb{R}^{n}$. Using the product rule, the differential of $y$ is$$ dy = dx^{T} x + x^{T} dx = 2 x^{T} dx $$
We can then set$$ dy = \frac{dy}{dx} dx = (\nabla_{x} y)^{T} dx = 2x^{T} dx $$where $dy/dx \in \mathbb{R}^{1 \times n}$ is called the derivative (a linear operator) and $\nabla_{x} y \in \mathbb{R}^{n}$ is called the gradient (a vector).
Now we can see $\nabla_{x} y = 2 x$.
If $x$ is complex, the complex derivative does not exist because $z \mapsto |z|^{2}$ is not a holomorphic function.
We can, however, instead consider the real derivatives with respect to the two components of $x$. Let $x = u + i v$. With this definition, $y$ is a real function of $u, v \in \mathbb{R}^{n}$ defined by$$ y = x^* x = (u + i v)^* (u + i v) = u^T u - v^T v $$Taking the differential$$ dy = 2 u^T du - 2 v^T dv = \frac{\partial y}{\partial u} du + \frac{\partial y}{\partial v} dv $$and therefore$$ \nabla_{u} y = 2 u \enspace , \qquad \nabla_{v} y = -2 v $$
For an introduction to matrix differential calculus, see the lecture of Geoff Gordon on YouTube or the paper on matrix derivatives of Mike Giles.
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