# Partial Derivatives and Physics meaning

What is physics meaning of higher partial derivatives, for example, the physics meaning for first derivatives is velocity and the second is acceleration, but what about

$$\frac{\partial^2 f}{\partial y\partial x} \quad \text{or} \quad \frac{\partial^2 f}{\partial x^2}$$

I'm looking for this question to get answer, I checked out the website but I don't get the answer.

It really depends on both the quantity you are taking the derivative of and the variable you are taking a derivative with respect to.

When you say

the physics meaning for first derivatives is velocity and the second is acceleration

what you actually mean is that the first derivative of position with respect to time is velocity, and the second derivative of position with respect to time is acceleration. There is in fact a name in physics for a higher derivative, namely the third derivative of position with respect to time, which is called jerk. This is the rate of change of acceleration, and is not a very commonly-used quantity. This is because in physics, quantities that connect different concepts are often more useful. E.g. velocity connects the motion of a body over time (kinematics) to the energy that the body has through the kinetic energy equation, acceleration connects kinematics to the force that caused this acceleration (through Newton's 2nd law), on the other hand, there are no commonly used concepts connected with jerk as far as I am aware of.

Similarly, while you can give a name to all derivatives of all quantities with respect to all the variables they depend on, not all of these will be concepts that are useful in physics.

$$\frac{\partial^2 f}{\partial x \partial y}, \frac{\partial^2 f}{\partial x^2}$$
If you are using the notation of $$f$$ meaning force and $$x$$ meaning position, the derivative of force with respect to position is not a generally useful quantity in physics, and the second derivative is also not. However, you can have the derivative of force with respect to area, which is the definition of pressure.
If you are interested in a particular $$f$$, $$x$$, and $$y$$, I can update my answer to try to elaborate more on specific derivatives.