You do need to take derivatives: twice w.r.t. x, and twice w.r.t t. I would re-write your u(x,t) for clarity. It's not clear, for example, exactly what's inside the argument of the second sin function. Use parentheses!
For the given Partial Differential Equation, show that the function u is a solution by plugging it into the Partial Differential Equation.
I think i need to find the derivative of u, however, I am unsure how to find a derivative with both du/dx and du/dt in the solution.