I know that that $$\dfrac{d\sqrt{x}}{dt} = \dfrac{d\sqrt{x}}{dx} \dfrac{dx}{dt}$$
In this equation there you only have 1 variable, namely $x$.
But why is the following correct?:
$$T = \frac{1}{2} m \left(v_{x}^2 + v_{y}^2 + v_{z}^2 \right)$$
$$\dfrac{dT}{dt} = m \left( v_{x} \dfrac{dv_{x}}{dt} + v_{y} \dfrac{dv_{y}}{dt} + v_{z} \dfrac{dv_{z}}{dt} \right)$$
How do you use the chain rule with this 3 variables and what is the mathematical proof for that?