# Leibniz notation - the good, the bad and the ugly

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

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

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

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.

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 $$\frac{d}{dx} \left( x^2 + x^3 \right)$$.

• 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 $\frac{df}{dx}(a)$? Oct 17, 2022 at 14:41
• @MathDoctor: yep. Oct 17, 2022 at 16:24