$dT/dx=0$ always true?

Question

In a Classical Mechanics book I found the assumption that for an arbitrary particle with constant mass in the Real line $dT/dx=0$, with T the Kinetic Energy i.e. $T=(m·dot x^2)/2$
My hypothesis is that the author used the following 'identity'
$$ddot x^2/dx=0$$
But solving the differential equation (correct me if I am wrong please) I get to $dot x=f(t)$
Which I think could wrong because it could be that $dot x=f(x,t)$ couldn't it?

Young Kindaichi · Answer

In classical Mechanics, where we work with Lagrangian $mathcal{L}(q,dot{q},t)$ which is a function of generalizing coordinate and velocity, We take velocity and coordinate to be independent variables. Why? Look here.
So kinetic energy which is
$$T=f(dot{q})$$
When we take its derivative with respect to $q$, it's turn out to be zero.
$$frac{dT}{dq}=0$$
As it doesn't depend on the generalized coordinate.

$dT/dx=0$ always true?

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