If $x:[0,a]\to \Bbb{R}^n$ is differentiable, what is $\frac{d}{dt}\Vert x(t)\Vert^2?$

Question

Let $\Vert\;\Vert$ be the Euclidean norm on $\Bbb{R}^n$. Let $x:[0,a]\to \Bbb{R}^n$ be differentiable. How do I define \begin{align}\frac{d}{dt}\Vert x(t)\Vert^2?\end{align} Please, I need help on this! A detailed answer would suit me. Thanks a lot!

Answer 1

In vector form $$\frac{d}{dt}\Vert x(t)\Vert^2=\frac{d}{dt}(x\cdot x)=(\dot{x}\cdot x)+(x\cdot \dot{x})=2\dot{x}\cdot x$$

Answer 2

If $x(t)=(x_1(t),...,x_n(t))$, then

$f(t)=||x(t)||^2=x_1^2(t)+...+x_n^2(t)$, hence

$f'(t)=2x_1(t)x'_1(t)+...+2x_n(t)x_n'(t)$.