Find the partial derivatives of the first order for $f(x,y,z) = z\sin(z^3y^x)$

Question

I have attempted the problem and would like to confirm my results and learn from the professionals.

Actions taken:

I started by differentiating the function with respect to x while treating y and z as constants. $$ \frac{{\partial f}}{{\partial x}} = z^4x\ln(y)\sin(z^3y^x) $$

Next, I proceeded to differentiate the function with respect to $y$ while treating $x$ and $z$ as constants. My result is:

$$ \frac{{\partial f}}{{\partial y}} = xz^4y^{x-1}\sin(z^3y^x) $$

3. Finally, I differentiated the function with respect to (z) while treating (x) and (y) as constants. My result is:

$$ \frac{{\partial f}}{{\partial z}} = \sin(z^3y^x) + 3z^3y^x\cos(z^3y^x) $$

I would appreciate it if the professionals on this platform could verify my calculations and provide any feedback or corrections if necessary. Thank you in advance for your assistance.