Find the total differential. \[ w=e^{7 y} \cos (5 x)+z^{4} \]

To find the total differential of the function \( w = e^{7y} \cos(5x) + z^4 \), we need to differentiate it partially with respect to each variable \( x \), \( y \), and \( z \). 1. The partial derivative with respect to \( x \): \[ \frac{\partial w}{\partial x} = e^{7y} (-5 \sin(5x)) = -5e^{7y} \sin(5x) \] 2. The partial derivative with respect to \( y \): \[ \frac{\partial w}{\partial y} = 7e^{7y} \cos(5x) \] 3. The partial derivative with respect to \( z \): \[ \frac{\partial w}{\partial z} = 4z^3 \] Now, we combine these partial derivatives to write the total differential: \[ dw = \frac{\partial w}{\partial x} dx + \frac{\partial w}{\partial y} dy + \frac{\partial w}{\partial z} dz \] Substituting the partial derivatives we calculated: \[ dw = (-5e^{7y} \sin(5x)) dx + (7e^{7y} \cos(5x)) dy + (4z^3) dz \] Thus, the total differential of \( w \) is: \[ dw = -5e^{7y} \sin(5x) \, dx + 7e^{7y} \cos(5x) \, dy + 4z^3 \, dz \]