When you take a partial derivative with respect to a variable, in this case x, the other variables are treated as constants and do not change.
∂z/∂x = ∂/∂x(e^x*cos(y))=>e^x*cos(y)
the derivative of e^x is itself e^x times the derivative of x which is one. e^x*1=e^x and cos(y) tags along for the ride, because it's a partial with respect to x, not y.
so the second derivative with respect to x will have the same answer.
∂^2z/∂x^2 = e^x*cos(y).
So now, to find the first partial with respect to y, e^x becomes a constant and the derivative of cos(y) is just -sin(y).
∂z/∂y = e^x*(-sin(y))
for the second partial with respect to y, we treat -e^x as a constant and derive sin(y), which is cos(y)
∂^2z/∂y^2 = -e^x*cos(y)
Now adding together we get ∂^2z/∂x^2 + ∂^2z/∂y^2 = e^x*cos(y) + (-e^x*cos(y)) = 0