Working with Conditions or Assumptions in Mathematica with boolean operators I have the following code:
$Assumptions = {x > 0}
b[x_] := x^2
b'[x] > 0

In my (very basic) understanding of Mathematica, this should give me me the Output True, but I get 2 x > 0. 
I also tried b[x_] := x^2 /; x > 0 and Assuming[x > 0, b'[x] > 0]. I've searched the mathematica help, but without success. What's my basic error and how do I get the desired output?

EDIT: The original question is answered, now I wanted to adapt this solution to two variables:
c[x_, y_] := x^2 + y
$Assumptions = {y > 0}
$Assumptions = {x > 0}
Simplify[c[x, y] > 0]

It follows the same logic as the first case, where I now get the desired output, but why not here? I realize that these are probably typical beginners questions, so if you could explain the logic to me or give me a hint where to read up upon this stuff? Neither the Mathematica help nor my university's (very short) guidebook are sufficient for my understanding.

Your first code
$Assumptions = {x > 0}
b[x_] := x^2
b'[x] > 0

works fine if you apply
 Simplify

to the result (2x > 0).
Edit: 
For completeness, I also add the answer of J.M in the comment to the second question.
 $Assumptions = {x > 0} 

overwrites 
  $Assumptions = {y > 0}. 

Try 
$Assumptions = x > 0 && y > 0.

