how long does it take to drive from las vegas to new york via grand canyon and yellow stone national park?

i want to go on holiday with my husband and children from las vegas to new york the scenic way and take in the grand canyon yellow stone national park Niagara falls along the way but im concerned about the duration as i need to book flights to suit the duration

Try 10 days.
Having driven it .lived it & hiked it I can tell you with 100% certainty that Las Vegas to NYC in anything under 4 days is a fast trip …and you guys want see something other than just what’s visible from the road.
A great routing from Las Vegas to Yellowstone looks like this
Pop this into mapquest or whatever exactly like this
LAS>GCN >Lees Ferry AZ.>Jacob Lake AZ > Colorado City AZ >Hurricane UT >Mount Carmel Junction UT
Stop
At Mt Carmel UT you meet up with US 89 N again and you can take that all the way into Jackson WY/Grand Teton NP into Yellowstone NP
Stop
Heading east…finally
Cody WY >Devils Tower WY> Rapid City SD
Across the Plains on I-90 to Madison WI
Stop
If you have no great urge to see the City of Chicago there are a couple of ferries that cross lake Michigan
Here’s one of them
http://www.ssbadger.com/content.aspx?Page=Schedule
Then
Manitowoc WI >Ludington MI >Windsor ON [I’m sending you to Niagara Falls starting on the Canadian side
Then
Niagara Falls NY >Syracuse NY > Scranton PA > East Stroudsburg PA >JFK
Have fun
Notes
Having one of these will come in handy
http://www.nps.gov/fees_passes.htm
Every summer there is huge motorcycle rally centered on Sturgis SD the result of which is that lodging can be hard to find for many a mile
http://www.sturgis.com/
On a more nerdy vein
Every year the EAA holds their annual fly-in at Oskosh WI and again lodging can be hard to find
http://www.airventure.org/
While Detroit MI is rapidly becoming a place that gives armpits a bad name there is one place in the area that is still truly amazing and should be visited if you’re in the area ; The Henry Ford Museum and Greenfield Village
http://www.hfmgv.org/