derivative dot(newton) notation $\frac{d}{dt} \dot{x}^2$

Question

I can't find how to derive $\dot{x}^2$

I know $\frac{d}{dt} \dot{x} = \ddot{x}$

My guess $\frac{d}{dt} \dot{x}^2 = 2\ddot{x}$, but I'm not sure.

Answer 1

Use the chain rule.

Remember that $\dot{x} = \frac{dx}{dt}.$ So $\dot{x}^2 = \left(\frac{dx}{dt}\right)^2.$ Now use the chain rule: the derivative is $2\dot{x} \ddot{x}$.