Find $\nabla f$ at the given point. \[ f(x, y, z)=x^{3}+y^{3}-4 z^{2}+z \ln x \] \[ \left.\nabla f\right|_{(1,2,3)}=\square \mathbf{i}+(\square) \mathbf{j}+(\square) \mathbf{k} \text { (Simplify your answers.) } \]

Step 1 ：Find the partial derivatives of the function: \(f(x, y, z) = x^{3} + y^{3} - 4z^{2} + z \ln x\)

Step 2 ：\(\frac{\partial f}{\partial x} = 3x^{2} + \frac{z}{x}\)

Step 3 ：\(\frac{\partial f}{\partial y} = 3y^{2}\)

Step 4 ：\(\frac{\partial f}{\partial z} = -8z + \ln x\)

Step 5 ：Evaluate the partial derivatives at the point (1, 2, 3):

Step 6 ：\(\left.\frac{\partial f}{\partial x}\right|_{(1,2,3)} = 3(1)^{2} + \frac{3}{1} = 6\)

Step 7 ：\(\left.\frac{\partial f}{\partial y}\right|_{(1,2,3)} = 3(2)^{2} = 12\)

Step 8 ：\(\left.\frac{\partial f}{\partial z}\right|_{(1,2,3)} = -8(3) + \ln 1 = -24\)

Step 9 ：\(\boxed{\nabla f|_{(1,2,3)} = 6\mathbf{i} + 12\mathbf{j} - 24\mathbf{k}}\)