Why does the mixed partial derivative of $ sin(x)+sin(y)=0$

Question

I wonder why and how it works. Please show me the steps, thanks

Answer 1

Let $f(x,y)=\sin(x)+\sin(y)$. Then $f_x(x,y)=\cos(x)$, which is independent of $y$. Therefore:

$f_{yx}(x,y)=0$.

Answer 2

When you differeniate wrt. $x$, you have $\cos x$, which is constant wrt. $y$.