Do I have enough time to make my connection train?

Simple answer: YES! will have enough time to change trains and continue the trip --::

Depends if your incoming train is on time, but if it is then you should be OK - Derby trains leave from platform 4, and if you're coming from the north or Manchester you will arrive on platform 5 or 1 on the same island platform. If you're coming from the London or Birmingham direction (though if you were I'd expect you to change trains at Stafford and Stoke rather than Crewe), or coming from north Wales you'll probably arrive on platforms 6 or 11 (if it's a through train to London) or 9 or 10 (if terminating in Crewe), or from south Wales on 6, 7, or 8; in any case, platforms 6 to 11 are one island platform over from 1-5 so you need to go up the stairs, over the bridge, and down to 1-5. If you're particularly unlucky, a few trains use platform 12 which is two island platforms over from 1-5.

If you use the lesser-used stairs at the south end of the platform, the bottom of the platform 1-5 staircase is at the end of platform 4, so. just turn left twice at the bottom of the stairs and your train should be on your left.

Crewe has a 10 minute connection time so it's likely to be long enough.

In the event you do miss your connection (as long as it's not your fault - eg the train was late) your ticket will be valid on the next train for ALL ticket types