Customizing math symbol "evaluated at" - Vertical bar growing up only below

Question

I'd like to customize the math symbol "evaluated at". If I use the code:

\documentclass[11pt]{article}
\usepackage{amsmath}

\begin{document}

\begin{equation*}
\frac{d}{dt} \left( i_{O_L} \Bigr|_{O_G} \right) =
\end{equation*}

\begin{equation*}
\frac{d}{dt} \left( i_{O_L} \Bigg\rvert_{O_G} \right) =
\end{equation*}

\end{document}

In the first case the vertical bar is too small. In the second case the vertical bar grows up in both directions up and down. I'm wondering if it's possible to grow up the bar only below. For example (i want a thin vertical bar, in fugure it is not so thin):

---

EDIT:

Thank you so much for your time.

Answer 1

Not too sure I've well ubderstood what you want, but I propose this:

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{diffcoeff, array, delarray}

\begin{document}

\begin{equation*}
\diffp*{\begin{array}[t]({@{}c@{\,}|@{\,}c@{}})O_L & \\ & O_T\end{array}}{t}
\end{equation*}

\begin{equation*}
\diffp*{\begin{array}[t]({@{}c@{\,}|@{\,}c@{}})O_L & \\[2ex] & O_T\end{array}}{t}
\end{equation*}

\end{document} 

Answer 2

Here's a modification of https://tex.stackexchange.com/a/458265/4427 where the rule only extends from the formula axis down, with a very large amount just for demonstration purposes.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\evalat}[2]{\mathpalette\eval@at{{#1}{#2}}}
\newcommand{\eval@at}[2]{\eval@@at#1#2}
\newcommand{\eval@@at}[3]{%
  #2\,
  {%
   \sbox0{$#1\left|#2\right|$}%
   \vrule height \fontdimen22\textfont2 depth \dimexpr\dp0+5ex\relax
  }_{\,#3}
}
\makeatother

\begin{document}

\[
\evalat{x}{x=5}
\qquad
\evalat{\frac{x+1}{x-1}}{x=2}
\]

\end{document}

Fix the depth by changing the value 5ex to something more sensible. Be advised that delimiters using \left and \right will grow as much in the upper as in the lower direction.