I don't have much time at the moment, but today I would like to the physical interpretation of the derivative (the title may be somewhat misleading, as physicists also use the geometric and analytic interpretation of the derivative).

We are often taught that the derivative of the function is the instantaneous rate of change, yet we are not given a reason as to why the slope of a tangent line is so. In this thread I will provide the reason that I use:

We will consider the most common rate of change for our purposes, that is velocity. While it is true that only constant velocities can be expressed by linear functions, y = mx + b, where m is velocity, it is also true that at any given point in time an object can have one and only one velocity. Therefore, at any given point in time an object's velocity can be expressed as a linear function (and of course if the object's velocity varies over a time interval the linear functions that describe its velocity also vary).

We have thus far concluded that an object's velocity at any given point in time can only be represented by a line, however we must also explain why a tangent line is used to complete our interpretation of the derivative (as opposed to merely the slope of any secant line):

the tangent line is chosen as it allows us to determine instantaneous velocity at its point of tangency on the basis that we can only clearly express on object's position with a tangent line (secant lines, which can cross at more than one point locally, cannot clearly identify an object's position).

This is a good method of interpreting the physical interpretation of the derivative for physicists and mathematicians alike.