Suppose we have:
Find the instantaneous rate of change of the area of a square with respect to its side when the side is $2\,\textrm{cm}$.
Solution.
Let $a$ be the side of the square. Then $A=a^2$ is the respective area and
$$
\dfrac{\textrm{d}A}{\textrm{d}a} = \dfrac{\textrm{d}}{\textrm{d}a}(a^2) = 2a.
$$
Finally,
$$
\left.\dfrac{\textrm{d}A}{\textrm{d}a}\right|_{a=2} = 4.
$$
Now, the question:
Is it $4\,\textrm{cm}$? I mean the units 🙂 Or it would be better to say $\textrm{cm}^2/\textrm{cm}$?
The later is the same as $\textrm{cm}$, but since we are asking about the rate of change in the area, maybe we should keep it as $\textrm{cm}^2/\textrm{cm}$?
Best Answer
Technically, the unit is simply $\mathrm{cm}$, yes.
However, for ease of understanding, I would say $\mathrm{cm}^2/\mathrm{cm}$ is better, as it gives a clear understanding of what is happening - that is we are giving the change of area (in $\mathrm{cm}^2$) per change of length (in $\mathrm{cm}$).