Find the total differential. \[ z=5 x^{7} y^{9} \]

To find the total differential \( dz \) of the function \( z=5 x^{7} y^{9} \), we need to compute the partial derivatives of \( z \) with respect to \( x \) and \( y \). First, we calculate the partial derivative with respect to \( x \): \[ \frac{\partial z}{\partial x} = 5 \cdot 7 x^{6} y^{9} = 35 x^{6} y^{9} \] Next, we calculate the partial derivative with respect to \( y \): \[ \frac{\partial z}{\partial y} = 5 \cdot 9 x^{7} y^{8} = 45 x^{7} y^{8} \] Now, we can write the total differential \( dz \): \[ dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy \] Substituting in our partial derivatives, we get: \[ dz = (35 x^{6} y^{9}) dx + (45 x^{7} y^{8}) dy \] Thus, the total differential is: \[ dz = 35 x^{6} y^{9} dx + 45 x^{7} y^{8} dy \]