# I What is the Result of this Partial Derivative

#### ecastro

What is the result of this kind of partial differentiation?

\begin{equation*}
\frac{\partial}{\partial x} \left(\frac{\partial x}{\partial t}\right)
\end{equation*}

Is it zero?

Thank you in advance.

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#### haruspex

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What is the result of this kind of partial differentiation?

\begin{equation*}
\frac{\partial}{\partial x} \left(\frac{\partial x}{\partial t}\right)
\end{equation*}

Is it zero?

Thank you in advance.
Out of context it means nothing. A partial derivative means changing the indicated variable while keeping some other variable(s) constant. Usually it is obvious what those other variables are. In a 3D coordinate system partial wrt one coordinate implies keeping the other two constant.
You need to provide a context for the expression.

#### ecastro

I apologize for the missing context. For example, $x$ signifies position and $t$ as time.

#### haruspex

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I apologize for the missing context. For example, $x$ signifies position and $t$ as time.
In that case I assume that partial wrt x means other spatial coordinates are held constant, but what is the significance of the partial wrt to t? What is being held constant there? I.e., why is it not just dx/dt?

Anyway, interpreting it as dx/dt:
Consider some line of particles or elastic thread along the x axis. If we take x as the location of some element at time t, we can ask how quickly it is moving along the x axis: dx/dt. The answer may be different for different points along the line, i.e. at different x values.
We could then ask how rapidly this velocity changes as we look along the line. This is the velocity gradient, $\frac d{dx}\frac{dx}{dt}$.

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