So I was going through the proof of a proposition which states that, 'If a solution of the nonhomogeneous wave equation exists, then it is unique.' It is the very first proof in the following pdf: The Cauchy Problem for the Nonhomogeneous Wave Equation
But, on page 2 of the above pdf, I did not understand how, while differentiating $v_t(x,t)$ again, or, while finding out $v_{tt}(x,t)$, they differentiated with respect to t, but wrote $F_x(...)$ in the next step?
Can you please explain how they did that?