Check for total differential?

Have a nice day,

I have a question regarding the following task:

Are these examples a total differential: z= sin(x^2+y^3)

z = ln(3x-2y)

I would have said that both are total differentials. First, I calculated dz. That means I first calculated a partial derivative with respect to x and then the partial derivative with respect to y. Using the example z = sin (x^2 + y^3), dz would be: dz = cos(x^2 + y^3) * 2x dx + cos(x^2 + y^3) * 3y^2 dy. Then I took another partial derivative, but the dx part with respect to y and the dy part with respect to x. If both derivatives produce the same result, which is the case in these two examples, then it is a total differential. However, I always did the proof with the integral, which means the initial function was already dz and I calculated the antiderivative. Can one calculate from the antiderivative as I have written here (double partial derivative) or can a total differential be proven in this way?