You've not said what time you're travelling!
For much of the day, on 5 April, there are 2 direct trains every hour from Newcastle to Derby. These normally depart at 25 and 40 minutes past each hour. The trains at 25 minutes past do not serve Wakefield Westgate, but those leaving at 40 minutes past do.
For example:
Newcastle 0940
Wakefield Westgate 1123
Derby 1222.
As you can see, Wakefield and Derby are about 1 hour apart by train. (The road distance is 67 miles).
To check details for the trains which match your requirements, go here:
http://www.nationalrail.co.uk/
Enter your journey details as Newcastle to Derby, with the relevant times and dates. When you get the results, click on 'See all details'. Then, for each train, you can click on 'See all stations called at for this journey leg', to check whether the train calls at Wakefield Westgate.
Chris