The Units of the Derivative
In the previous activity we used the derivative to identify the maximum height of the incoming missile. In short: we found the time when the derivative was equal to 0, and with some deductive reasoning, we deduced that this time was also the time that the missile achieved its maximum height. You may have wondered: what are the units of
g'(x)?
The answer is surprisingly simple: since the units of g(x) are meters, and the units of x are seconds, the units of g'(x) are meters per second.
This is a fact that applies to all derivatives:
Fact: For any function f(x) which is model of a real world phenomenon, if the units of f(x) are FARTs and the units of x are PERSONs, then the units of the derivative f'(x) are FARTs per PERSON. Of course, you can replace FARTs and PERSONs with any other physical quantities.
Sometimes the units of f(x) are called the units of the dependent variable, and the units of x are called the units of the independent variable. In which case, we'd say the units fo the derivative are:
The reason why the units of the derivative are always like this is because of the definition of what the derivative f'(x) is: a function that keeps track of the slopes of the tangent lines of f(x). And what are the units of the tangent line? Answer: The units of the rise divided by the units of the run. And these are precisely the units of the dependent and independent variables, respectively.
With that out of the way, move onward to begin studying an impressive sequence of shortcuts for calculating derivatives.