Partial derivative of position in respect to velocity

Question

Out of interest What is the derivative of $\frac{\partial x(t)}{ \partial v(t)}$

Answer 1

Let me note that, since both $x$ and $v$ are functions of a single variable, we are dealing here not with partial derivatives but with ordinary derivatives which are equivalent to the corresponding differentials (partial derivatives are not). This may seem like a simple change of notation $$ \frac{\partial x(t)}{\partial v(t)} \longrightarrow \frac{d x(t)}{d v(t)}, $$ but mathematically these are two different things. Importantly, ordinary derivative is actually a true ratio of two differentials, $dx = \dot{x}(t)dt, dv = \dot{v}dt$, while a partial derivative is just a symbol.

Now we can consider $x$ as a function of $v$: $x = f(v)$, which is given to us in a parametric form, with $t$ being the parameter. The ordinary derivative of this function is then just the ratio of the two differentials, i.e. $$ \frac{dx}{dv} = \frac{\dot{x}(t)}{\dot{v}(t)} $$ the answer is actually again given in a parametric form. If one wants it as a function of $v$; one needs to invert the function $v(t)$ and substitute it in the expression above. In other words $$ f'(v) = \frac{\dot{x}(t)}{\dot{v}(t)}|_{v(t)=v}. $$