ThmDex – An index of mathematical definitions, results, and conjectures.

Derivative function for euclidean real self-dot product function

Formulation 0

Let $f : \mathbb{R}^{N \times 1} \to \mathbb{R}$ be a D4364: Real function such that

(i)	\begin{equation} f(x) = x^T x \end{equation}

Then \begin{equation} f'(x) = 2 x^T \end{equation}

Formulation 1

Let $f : \mathbb{R}^{N \times 1} \to \mathbb{R}$ be a D4364: Real function such that

(i)	\begin{equation} f(x) = \Vert x \Vert^2_2 \end{equation}

Then \begin{equation} f'(x) = 2 x^T \end{equation}

Proofs

Proof 0

Let $f : \mathbb{R}^{N \times 1} \to \mathbb{R}$ be a D4364: Real function such that

(i)	\begin{equation} f(x) = x^T x \end{equation}

This result is a particular case of R4909: Derivative for euclidean real self-dot product function. $\square$