According to the book "Jacob Moak of New Scotland and his descendants 1720 - 1990", it would appear that William got the Van Der Hyden part of his name from one of his great grandmothers. (book published in Albany, New York in 1990 by Martha Slingerland and Grace Moak)
Here's the info as written in the book:
William VanderHyden Moak - b 9 June 1877 - lived near Chicago; no children
William's Parents:
Henry P. Bradt Moak - b. 14 July 1849, d. 1932 (Bethlehem Cemetery)
Mary M. Albright - b. 2 May 1843, d. 1933
married 12 Aug 1873
William's Grandparents: (Henry P. Bradt Moak's parents)
Nicholas B. Moak - b. 7 April 1824, d. 1903
Magdalen Bradt - b. 30 June 1830, d. 1906
married 24 Oct 1848
William's Great grandparents:
Baltis N. Moak & Hannah Radliff (Nicholas's parents)
Henry Peter Bradt & Magdalen VanderHyden (Magdalen's parents)
My 3x great grandmother was a Moak but a different branch than this line.
Hope this helps,
Pat