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First timer here. I would like to know the correct ways of writing and interpreting derivatives written using leibniz notation. I have a sample of 8 symbols below. For each one of them I would like to know:

  • whether it is incorrect notation
  • if it is not incorrect whether there is a better way to write the same
  • what function is the derivative operator acting on

$$\begin{array}{rrrr} \frac{\mathrm{d}x^2}{\mathrm{d}x}, & \frac{\mathrm{d}kx^2}{\mathrm{d}x}, & \frac{\mathrm{d}x^2+x^3}{\mathrm{d}x}, & \frac{\mathrm{d}\left(x^2+x^3\right)}{\mathrm{d}x},\\ \frac{\mathrm{d}}{\mathrm{d}x}x^2, & \frac{\mathrm{d}}{\mathrm{d}x}kx^2, & \frac{\mathrm{d}}{\mathrm{d}x}x^2+x^3, & \frac{\mathrm{d}}{\mathrm{d}x}\left(x^2+x^3\right) \end{array} $$

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    $\begingroup$ If you can't tell which function the derivative operator is acting on, it's bad notation. This is why you should bracket sums. $\endgroup$
    – J.G.
    Sep 14, 2022 at 16:49
  • $\begingroup$ i appreciate your point about sums. does this concern also extend to products? e.g. in the case (d/dx)x^2(x+1) $\endgroup$
    – khany
    Sep 18, 2022 at 3:41
  • $\begingroup$ Yes, it does, as that's also unclear. $\endgroup$
    – J.G.
    Sep 18, 2022 at 6:07

2 Answers 2

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Of course notation is not completely unambiguous. But here is how I would interpret your examples.

  1. $\frac{\mathrm{d}x^2}{\mathrm{d}x}$ should be differential of $x^2$ wrt $x$. Hence it should equal $2x$.

  2. $\frac{\mathrm{d}kx^2}{\mathrm{d}x}$. I do not know what $k$ is supposed to be. I assume it is some constant, in this case I calculate this to equal $2kx$, the differential of $kx^2$ with respect to $x$. Although I suppose one could argue this equals $0$, interpreting the nominator as $(\mathrm d k)x^2$, and the differential of constants vanishes.

  3. $\frac{\mathrm{d}x^2+x^3}{\mathrm{d}x}$. I'd say this doesn't make sense. To me $\mathrm d x$ should be a placeholder for something very small, and as $\mathrm d$ is not applied to $x^3$ (I'd think that addition should bind weaker than $\mathrm d$), I wouldn't know how to interpret $\frac {x^3}{\mathrm dx}$.

  4. $\frac{\mathrm{d}\left(x^2+x^3\right)}{\mathrm{d}x}$ is just $2x + 3x^2$

  5. $\frac{\mathrm{d}}{\mathrm{d}x}x^2$ This is the same as 1. to me.

  6. $\frac{\mathrm{d}}{\mathrm{d}x}kx^2$ is the same as in 2.

  7. $\frac{\mathrm{d}}{\mathrm{d}x}x^2+x^3$ is just $2x + x^3$.

  8. $\frac{\mathrm{d}}{\mathrm{d}x} \left(x^2+x^3\right)$ should be $2x + 3x^2$.

The pattern is that $\mathrm d$ binds more or less like multiplication with a variable.

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I would personally avoid every piece of notation in the top row. Most of it is understandable, but using $dx^2$ like this produces a really unfortunate collision with the second derivative notation $\frac{d^2 f}{dx^2}$ and I think this collision is confusing enough to justify completely avoiding using notation like $\frac{dx^2}{dx}$; e.g. if you find yourself in a position to teach someone calculus someday you don't want them asking whether $\frac{d^2 f}{dx^2} \frac{dx^2}{dx} = \frac{d^2 f}{dx}$, which doesn't make any sense but would be an understandable question, wouldn't it?

The worst piece of notation in the top row is $\frac{dx^2 + x^3}{dx}$; it's ambiguous whether you intended to differentiate $x^3$ and if you didn't then $\frac{x^3}{dx}$ doesn't make any sense. After all, we should have $\frac{dx^2 + x^3}{dx} = \frac{dx^2}{dx} + \frac{x^3}{dx}$, right?

Similarly, most of the second row is fine except for $\frac{d}{dx} x^2 + x^3$, which makes it too ambiguous whether you intended to differentiate $x^3$; I don't know that I'd call it "incorrect" but I'd call it bad style. If you really meant to differentiate $x^2$ only you should write $\left( \frac{d}{dx} x^2 \right) + x^3$. If you meant to differentiate both you should write $\frac{d}{dx} \left( x^2 + x^3 \right)$.

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  • $\begingroup$ Hi Qiaochu, I have a quick follow up question to your answer: for the derivative of $f$ at a point $a$ we write $f'(a)$, could we also write this using Leibniz notation as $\frac{df}{dx}(a)$? $\endgroup$ Oct 17, 2022 at 14:41
  • $\begingroup$ @MathDoctor: yep. $\endgroup$ Oct 17, 2022 at 16:24

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