How do I determine the units for something like $da/dx$?

Question

I have the following statement:

If the units for $x$ are feet and the units for $a(x)$ are pounds per foot, what are the units for $da/dx$?

The answer is $\frac{\text{pounds}}{\text{feet}^2}$. Why is that? From what I understand, $da$ is in units of $\text{pounds}$ because that's the difference for some small delta value for $a(x)$. I don't understand how $dx$ gives $\text{feet}^2$. Unless that's the antiderivative.

Answer 1

If the units for $x$ are feet and the units for $a(x)$ are pounds per foot the units for $\frac{da}{dx}$ are $$\frac{\text{pounds}}{\text{feet}^2}$$

Indeed $\frac{da}{dx}$ is the limit for $\Delta x\to 0$ of

$$\frac{\Delta a}{\Delta x}$$

which units are $\frac{\text{pounds}}{\text{feet}^2}$.

EG

Think to acceleration

$$a=\frac{dv}{dt}=\lim_{\Delta t\to 0} \frac{\Delta v}{\Delta t}\quad \left[\frac{\frac{m}{s}}{s}\right]=\left[\frac{m}{s^2}\right]$$

Answer 2

The reason is that $da/dx$ is the limit of a ratio, so it would have the same units as the ratio $a/x$.

Answer 3

You have $[a]=lb/ft$ and $[x]=ft$. An infinitesimal change of $a$, i.e. $da$, would have the same units as $a$, so $[da]=lb/ft$ (analogy: a fragment of an orange is still made of orange material). The same applies for $dx$: $[dx]=ft$. Then,

$$\left[\frac{da}{dx}\right]=\frac{[da]}{[dx]}=\frac{lb/ft}{ft}={lb/ft^2}$$