# Find a function M

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?

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.