what is physical meaning of this partial derivative?

what is physical meaning of this partial derivative: $$\frac {\partial p_x}{\partial x}$$ i know how do i solve it when the case is just derivative but partial derivative is a bit Hectic!.

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This question is lacking some context. A partial derivative is nothing but a normal derivative with the other variables taken as constants. What the physical meaning is depends on the function $p_x$. – Alexander Jun 16 '12 at 0:31
If $p_x$ is momentum then the derivative is the spatial gradient along the `x`-axis. – ja72 Jun 16 '12 at 1:56

3 Answers

If $p_x$ is taken to mean momentum in the $x$ direction, $\frac{\partial p_x}{\partial x}$ simply means that momentum in the $x$ direction changes as you move along $x$. So for an object with a given mass, $\frac{\partial p_x}{\partial x}$ describes how its $x$ velocity changes as the object moves along $x$. In everyday life, that would just be called an acceleration or deceleration, such as braking a car that is moving east, if you call east $x$.

Where partials get more complicated is that you can have several such things going on at once. More specifically, you can have other independent accelerations or decelerations going on in $y$ and $z$ (think roller coasters). Finally, for a partial derivative that is expressed in terms of momentum instead of velocity, you cannot always assume the mass is constant, so you also cannot always assume that the effects are equivalent to velocity expressions.

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The partial derivative of a function $f(x_1,x_1,x_3,..)$ at a point $(k,x_2,x_3,...)$, e.g. with respect to $x_1$, is the slope of the tangent line to the curve obtained by the intersection of the plane $x_1=k$, with the plot of $f(x_1,x_2,...)$.

The partial derivative also gives the rate of change with respect to the quantity being changed. Your partial derivative would give the rate of change of the momentum in the $x$ direction with respect to change in $x$, it essentially is a measure of how the velocity along the $x$ axis, if mass is constant changes, as you move along $x$, i.e. the change in a quantity along the direction you are moving in.

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A partial derivative basically tells you how a function changes if I change just one of many variables it may depend on, while keeping all other variables constant.

On the other hand, a total derivative tells you the "TOTAL" information about the function. It describes how a function changes if all (or some) variables are changed together.

When a function depends only on one variable then ALL = one variable. In this case, a partial derivative is the same as the total derivative.

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