I want to know what is the meaning of dσ/du in this solution?
For same shearing stress if absolute value of $\sigma_x$-$\sigma_y$ is larger, points on Mohr circle are wider apart and $R$ is larger thus maximum shear stress is larger. Variable $u$ is introduced to represent this connection and derivative was tried to obtain solution but it failed and solution was obtained using geometric properties of Mohr's circle. You should try to draw circle for better understanding.
This is not a direct answer to your question, but help you to remember what you have learnt at school.
What is the purpose of differentiation in calculus?
What is the "Chain Rule" and its usage in differentiation calculus?
Review Mohr's Circle for Plane Stresses.
If you have any problem in the effort of reviewing the topics stated above, come back to ask the specific question with your effort indicated.