d/dx(int(sqrt(t+sqrt(t)),t,0,tanx)))

This uses what is known as Leibnitz's rule for differentiation of integrals, which in its general form, where both limits are functions of the variable with respect to which everything is being differentiated, is:

In the original question here only the 2nd term on the RHS above is non-zero

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Yes, d/dx(int(sqrt(t+sqrt(t)),t,0,tanx)))  Your Welcome.

Your meaning is not clear 

What are you integrating with respect to ?

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Mmm

Another for me to think about - Thanks Alan  

Could you try and explain this a bit please Alan.   

How did you go from the first step to the second step Alan,

Is it magic ??

NOTE: much of mathematics is magic you know  

Congratulations JewishNerd.

I think that you have taken ownership of the ugliest icon imaginable.

What is that thing??     LOL

( I hope it is not your pet cat. )

Thanks Alan :))