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I have this working example:

\documentclass[letterpaper,12pt]{article}

\usepackage{hyperref}
\usepackage{amsmath}


\begin{document}


\begin{align}
\frac{d}{dx}&(\sin^{-1} x)=\frac{1}{\sqrt{1-x^2}}   &   \frac{d}{dx}&(\cos^{-1} x)=-\frac{1}{\sqrt{1-x^2}}  \
\frac{d}{dx}&(\tan^{-1} x)=\frac{1}{1-x^2}  &   \frac{d}{dx}&(\cot^{-1} x)=-\frac{1}{1-x^2} \
\frac{d}{dx}&(\sec^{-1} x)=\frac{1}{ x  \sqrt{x^2-1}    &   \frac{d}{dx}(\csc^{-1} x)=-\frac{1}{x  \sqrt{x^2-1}}
\end{align}


\end{document}

Compiling gives this error

Runaway argument?
 \frac {d}{dx}&(\sin ^{-1} x)=\frac {1}{\sqrt {1-x^2}} & \frac {d}{dx\ETC.
! Paragraph ended before \align* was complete.
<to be read again>
                   \par
l.13

I can't spot any errors with the align.

Shrimpram
  • 1
  • 5
    You miss a closing } in the third line of the equation. Use \frac{d}{dx}&(\sec^{-1} x)=\frac{1}{ x \sqrt{x^2-1}} & \frac{d}{dx}(\csc^{-1} x)=-\frac{1}{x \sqrt{x^2-1}}. –  Nov 03 '20 at 07:55
  • The alignment points and differential ds are also not entirely correct. Use \newcommand{\diff}{\mathop{}\!\mathrm{d}} \begin{align*} \frac{\diff}{\diff x}(\sin^{-1} x)&=\frac{1}{\sqrt{1-x^2}} & \frac{\diff}{\diff x}(\cos^{-1} x)&=-\frac{1}{\sqrt{1-x^2}} \\ \frac{\diff}{\diff x}(\tan^{-1} x)&=\frac{1}{1-x^2} & \frac{\diff}{\diff x}(\cot^{-1} x)&=-\frac{1}{1-x^2} \\ \frac{\diff}{\diff x}(\sec^{-1} x)&=\frac{1}{ x \sqrt{x^2-1}} & \frac{\diff}{\diff x}(\csc^{-1} x)&=-\frac{1}{x \sqrt{x^2-1}} \end{align*} –  Nov 03 '20 at 07:58
  • @anonymous what exactly is different about the \diff command you use and the align points. Sorry, I'm fairly new to latex, just trying to figure out how you solved the issue so I don't have to ask next time =) – Shrimpram Nov 03 '20 at 08:08
  • The error is from the missing }. As for the alignment points, it is a common practice to align at the equality signs. As for the differential, this is an operator and is usually typeset in the same font as other operators, namely upright. (Already Gauss and Euler used the same fonts for functions like sin and the differential d, but rumors say that they were not using LaTeX for some reason. ;-) –  Nov 03 '20 at 08:11
  • @anonymous A large number of mathematicians use the italic “d”. I suspect it's a vast majority. Anybody is free to do as their field is used to. If the OP uses an italic “d”, it's pointless to teach them differently. If you're so fussy, then why don't you mention that \sin^{-1} is wrong according to ISO regulations? – egreg Nov 03 '20 at 09:24
  • By the way, you can easily check that Euler never used an upright “d” for the differential (and italic for the sine, too). See this example – egreg Nov 03 '20 at 09:40
  • I am not "fussy". 2. There is a clear ISO convention. 3. The vast majority of those (mathematicians and other scientists) who actively think about how to typeset the differential typeset in the font in which they typeset functions, which is usually upright. 4. As one can see, in Euler's article they used the same font for the sine as for the differential. But this is anyway not very conclusive as Euler didn't typeset it himself. 5. It is common to denote the inverse of a function by $f^{-1}$.
    • –  Nov 03 '20 at 14:12
  • Remark: Readers coming here may be able to find their answer in Paragraph ended before \align was complete - TeX - LaTeX Stack Exchange. – user202729 Oct 09 '22 at 05:05