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Let $x(t)$ be a real valued vector. Can you find a function M such that

$\dot{M}=\frac{\text{d}M}{\text{d}t}=\dot{x}^T\dot{x}$.

I have tried

$M=\dot{x}^Tx, M=x^Tx$

and many more which don't work. I know that if $x(t)$ is continuous then so will

$\dot{M}=\dot{x}^T\dot{x}$

be and thus M will exist. But how do i find this M?

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Impossible a general formula even in dimension 1. In the case, you are asking for $$\int(x'(t))^2\,dt$$ and no such formula exists. In concrete cases maybe you can do the integral.

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