What is the unit of this "Rate of Change"?

Question

This is the plot of the function $f(x) = 0.9x$.

So, the rate of change of $y$ with respect to $x$ is $0.9$.

My question is, what is the unit of this Rate of Change?

$0.9$ per $x$?

Answer 1

Well, it really depends on context. In a very pure sense, the slope does not have units in this context. In a more vernacular sense, it's common to have things like $$y=\text{# of meters travelled}$$ $$x=\text{# of seconds passed}$$ Then write $y=2x$ as a relationship. Then, the units of slope would be $\text{meters/second}$, meaning that two meters pass (on the $y$-axis) for every one second on the $x$-axis. So, the units here are the units of the $y$-axis divided by those of the $x$-axis.

However, it's worth noting that this relationship might be better written as $$f(x\text{ seconds})=2x\text{ meters}$$ where you recognize that the function $f$ takes in a duration and spits out a length - and that the units are not an intrinsic quality of the function. In this sense, you could justifiably write for the same function that $$f(x'\text{ minutes})=120x'\text{ meters}.$$ An equivalent way to write this function is as follows: $$f(t)=(2\text{ meters/second})\cdot t$$ where you note that if you plug in $x\text{ seconds}$ for $t$, you get $2x\text{ meters}$. This last point of view shows that the slope is just the coefficient multiplied by the input - and it is very often desirable to treat these slopes as intrinsically having units which are part of the value, not an interpretation thereof.

---

In an abstract setting (or a setting where the input and output are both distances or both times or generally both the same), the slope does not have units. It is just $0.9$.

Answer 2

If you would put units for $x$ and $y$, say time in seconds for $x$ and distance in meters for $y$, the slope is in meters/second. So in the general case, the unit for the rate of change is "units of y divided by units of x". You can use "per" instead of "divided by".