Should I be worried about putting brackets around derivatives when I'm using chain rule?

Question

$$\frac{d}{d(2\cot\theta)}(\tan^{-1}2\cot\theta)\cdot\frac{d}{d(\cot\theta)}(2\cot\theta)\cdot\frac{d}{d\theta}(\cot\theta)\cdot\frac{d}{dx}\theta\tag{1}$$

$$\left(\frac{d}{d(2\cot\theta)}(\tan^{-1}2\cot\theta)\right)\cdot\left(\frac{d}{d(\cot\theta)}(2\cot\theta)\right)\cdot\left(\frac{d}{d\theta}(\cot\theta)\right)\cdot\frac{d}{dx}\theta\tag{2}$$

$$4\sin2x\cos2x\tag{3}$$

$$4(\sin2x)(\cos2x)\tag{4}$$

$(2)$ is completely unambiguous. No one can misinterpret anything. However, in $(1)$, one can misinterpret, for example, $\frac{d}{d(2\cot\theta)}(\tan^{-1}2\cot\theta)\cdot\frac{d}{d(\cot\theta)}(2\cot\theta)$ as meaning $\frac{d}{d(2\cot\theta)}\left[(\tan^{-1}2\cot\theta)\cdot\frac{d}{d(\cot\theta)}(2\cot\theta)\right]$ instead of $\left(\frac{d}{d(2\cot\theta)}(\tan^{-1}2\cot\theta)\right)\cdot\left(\frac{d}{d(\cot\theta)}(2\cot\theta)\right)$.

Similarly, $(4)$ is completely unambiguous. No one can misinterpret anything. However, in $(3)$, one can misinterpret $\sin2x\cos2x$ as meaning $\sin(2x\cos2x)$ instead of $(\sin2x)(\cos2x)$.

Is this a minor thing? Should I be worried about putting brackets obsessively? I experience so much anxiety now while doing math.

Clarity > brevity. In fact, brevity often results in ambiguity, which wastes more time than putting a few more symbols (or words) which would makes everything unambiguous. So I prefer $(2)$ and $(4)$ over $(1)$ and $(3).$ — Adam Rubinson, May 15 '22 at 13:42
@ryang I didn't see $(2)$ written anywhere. My book uses a different version of $(1)$, though:
$$\frac{d(\tan^{-1}2\cot\theta)}{d(2\cot\theta)}\cdot\frac{d(2\cot\theta)}{d(\cot\theta)}\cdot\frac{d(\cot\theta)}{d\theta}\cdot\frac{d\theta}{dx}\tag{*}$$ — tryingtobeastoic, May 15 '22 at 15:13
@ryang Now that I think about it, is $(*)$ the exact same as writing $(1)$? — tryingtobeastoic, May 15 '22 at 15:16
@ryang You're right. arctan is still ambiguous in (2), and (4) is also ambiguous still. Aaaaaaaaaaaaaaa — tryingtobeastoic, May 16 '22 at 02:58

score 2 · Accepted Answer · answered May 15 '22 at 19:23

2

Just as spaces are important in separating words, so they are in mathematical notation. Thus $\sin2x$ means $\sin(2x)$, and $\sin2x\cos2x$ means $(\sin2x)(\cos2x)$. If we really mean $\sin(2x\cos2x)$, then we have to add the parentheses to override the spacing convention. To override the usual convention that multiplication (denoted by juxtaposition) operates before other operations, we can use a centered dot. Thus $\sin2\cdot x$ means $(\sin2)x$ rather than $\sin2x$.

The spacing convention implies that the notation $\tan^{-1}2\cot\theta$ means $(\tan^{-1}2)\cot\theta$. If we intend $\tan^{-1}(2\cot\theta)$, then the parentheses are necessary.

The dots in your expression #1 make it unambiguous, and there is no need to add the (big) parentheses as in expression #2. Also, the parentheses in $\dfrac{\mathrm d}{\mathrm d\theta}(\cot\theta)$ are redundant, but they are not redundant in $\dfrac{\mathrm d}{\mathrm d\theta}(2\cot\theta)$. Thus $\dfrac{\mathrm d}{\mathrm d\theta}\cot\theta=-\csc^2\theta$, while $\dfrac{\mathrm d}{\mathrm d\theta}2\cot\theta=\dfrac{\mathrm d}{\mathrm d\theta}2\cdot\cot\theta=0$.

answered May 15 '22 at 19:23

John Bentin

18,454

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I'm having some trouble understanding what the spacing convention is. In $\sin 2x$, we interpreted it as $\sin(2x)$, probably because there is a single space between $\sin$ and $2x$, and there is no space between $2$ and $x$ in $2x$. In $\tan^{-1}2\cot\theta$, there is a single space between $\tan^{-1}$ and $2\cot\theta$, and there is no space between $2$ and $\cot\theta$ in $2\cot\theta$, just like in the case of $\sin 2x$; why is it then interpret-able as $(\tan^{-1}2)\cot\theta$ instead of $\tan^{-1}(2\cot\theta)$ according to the spacing convention? – tryingtobeastoic May 16 '22 at 03:16
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@tryingtobeastoic 1. Mathematical writing, very much like natural language, is not a formal language, and conventions (spacing, product/centred dots, etc.) are not rigorous systems, but can be combined and meant to complement judgement and unspoken conventions (and the surrounding context). I'm confident that you can typically detect expressions that are actually ambiguous (rather than merely technically ambiguous like my above contrived $(\sin2)x$ example). That $(\sin2)x$ ambiguity was tricky to detect precisely because nobody would read it that way. – ryang May 16 '22 at 04:39
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@tryingtobeastoic 2. "In $\tan^{-1}2\cot\theta,$ there is a single space between $\tan^{-1}$ and $2\cot\theta,$ and there is no space between $2$ and $\cotθ$ in $2\cotθ$" $;$ But there *is* a (thin) space between $2$ and $\cotθ;$ it is automatically inserted by mathjax, and is why $\sin\theta\tan(x)$ and $\sin \theta \tan (x)$ have exactly the same output. – ryang May 16 '22 at 04:54

Should I be worried about putting brackets around derivatives when I'm using chain rule?

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