Find the total differential. \[ z=\frac{x^{4}}{y^{3}} \]

To find the total differential \( dz \) of the function \( z = \frac{x^4}{y^3} \), we first compute the partial derivatives with respect to \( x \) and \( y \). 1. **Partial derivative with respect to \( x \)**: \[ \frac{\partial z}{\partial x} = \frac{4x^3}{y^3} \] 2. **Partial derivative with respect to \( y \)**: \[ \frac{\partial z}{\partial y} = -\frac{3x^4}{y^4} \] Now, the total differential \( dz \) is given by: \[ dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy \] Substituting the partial derivatives we computed: \[ dz = \frac{4x^3}{y^3} dx - \frac{3x^4}{y^4} dy \] So the total differential is: \[ dz = \frac{4x^3}{y^3} dx - \frac{3x^4}{y^4} dy \]