What is the unit of this "Rate of Change"?
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 $x$?
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$\begingroup$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".
$\endgroup$ $\begingroup$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 $0.9$.
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