According to symbolic MATLAB and WolframAlpha,
$\frac{\partial x(t)}{\partial x} = 0, \frac{\partial x}{\partial x} = 1$
I came across this while trying to figure out how to do:
$\frac {\partial} {\partial x} \int \dot x dt$
How can this be?
$\endgroup$1 Answer
$\begingroup$Think about $\frac{dx}{dx}$ as $\frac{d}{dx}(x)$ which means the derivative of the function $x$ (taking $x$ to be the variable--from the denominator of the fraction). Whereas $\frac{dx(t)}{dx}$ means $\frac{d}{dx}(x(t))$ so the derivative with respect to $x$ of some function of $t$. A function of $t$ has no variable $x$, so appears as a constant. Hence, the derivative is $0$. For example, think $\frac{d}{dx}(t^2)=0$... here $x(t)=t^2$.
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