\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
W(f_1,\dotsc,f_g) := \text{det}
\begin{pmatrix}
f_1 & f_2 & \dots & f_g \\
f_1' & f_2' & \dots & f_g' \\ % <---
. & . & & . \\
. & . & & . \\
. & . & & . \\
f_1^{(g-1)} & f_2^{(g-1)} & \dots & f_g^{(g-1)}
\end{pmatrix}
\end{equation}
or better
\begin{equation}
W(f_1,\dotsc,f_g) := \det
\begin{pmatrix}
f_1 & f_2 & \dots & f_g \\
f_1' & f_2' & \dots & f_g' \\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(g-1)} & f_2^{(g-1)} & \dots & f_g^{(g-1)}
\end{pmatrix}
\end{equation}
\end{document}

Addendum:
To some people prefer to use instead := different symbols for definition. Discussion and different symbols about this symbols you can find in How to typeset := correctly. For example \coloneqq from the package mathtools or \colonequals from the package colonequals, etc. Differences between are subtle, however for people with with a keen sense of aesthetics they are important. For comparison observe the case of use of \colonequals:
\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{equation}
W(f_1,\dotsc,f_g) \coloneqq \det\begin{pmatrix}
f_1 & f_2 & \dots & f_g \\
f_1' & f_2' & \dots & f_g' \\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(g-1)} & f_2^{(g-1)} & \dots & f_g^{(g-1)}
\end{pmatrix}
\end{equation}
\end{document}

Further possibility is use the unicode-math with xelatex˙orlualatexengine. This fonts define\coloneq` which gives:

@Sebastiano, thank you for pointing me on this detail. Personally I very rare if ever use (due to my professional background) symbols for equal by definition :-)
f_1^{\prime}instead off_1^'. – Cragfelt Nov 29 '19 at 12:56\detinstead of\text{det}. – egreg Nov 29 '19 at 13:03