Notation of derivatives

Question

Small question, If $u=dx/dt$, and we have $a=du/dt$, is this equal to $a=du/dt=dx/dt^2=(1/dt)(dx/dt)=u/dt$?

$a=du/dt=d^\color{red}2x/dt^2$ – J. W. Tanner Apr 28 '20 at 23:34 — J. W. Tanner, Apr 28 '20 at 23:34

score 1 · Answer 1 · answered Apr 28 '20 at 23:36

1

If $u=\dfrac{dx}{dt}$, then $\dfrac{du}{dt} = \dfrac{d^2x}{dt^2}$ (note the different position of the squares).

However, neither $\dfrac{1}{dt}$ nor $\dfrac{u}{dt}$ make sense in standard calculus. That’s because it’s not really a fraction

answered Apr 28 '20 at 23:36

Arturo Magidin

398,050

score 0 · Answer 2 · answered Apr 28 '20 at 23:35

No, absolutely not. That's not how the derivative notation works, and is why it might be preferable to use the $f'$ sort of notation.

So, we have that $u = x'$ and $a = u'$. Then this means that $a = (x')' = x''$. Thus, in your more familiar notation,

$$a = \frac{d^2 x}{dt^2}$$

In general, it is bad practice to treat $df/dt$ as a fraction. While it often makes for a convenient abuse of notation, and in some cases there is a rigor behind it, it's still generally unfounded. Treat $d/dt$ or any other sort of operator as an entire entity on its own, not a fraction.

Notation of derivatives

2 Answers2