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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?

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$v_x,v_y,v_z$ these are three dependent variables. There is only one independent variable $t$. Infact $v_x$ is a function of both position $x$ and time $t$ but the position $x$ is itself a function of $t$ so all the components of $v$ are just a function of $t$.

If there were some other independent variable $t'$ then we would talk about partial derivatives and the multivariable chain rule.

About the proof: the usual proof of chain rule is valid. $v_x$ is just a function of $t$. There might be some other functions of $t$ like acceleration but the chain rule is still valid.

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