91Ó°ÊÓ

In order to find the differential of the function \(w = f(x, y, z)\), we first need to find its partial derivatives. By definition, the partial derivative is the rate at which the function changes with respect to a specific variable, holding the others constant. These derivatives are denoted as follows: \(\frac{\partial w}{\partial x} = \frac{\partial f(x, y, z)}{\partial x}\) \(\frac{\partial w}{\partial y} = \frac{\partial f(x, y, z)}{\partial y}\) \(\frac{\partial w}{\partial z} = \frac{\partial f(x, y, z)}{\partial z}\) Since we do not have the specific function \(f(x, y, z)\), we will keep these partial derivatives in a general form as \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), and \(\frac{\partial w}{\partial z}\).

Now, we need to incorporate these partial derivatives into the definition of the differential. The total differential \(dw\) for the function \(f(x, y, z)\) is given by the following formula: \(dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy + \frac{\partial w}{\partial z}dz\)

Having calculated the partial derivatives and expressed the differential with their help, we finally write down the differential of the function \(w = f(x, y, z)\): \(dw = \frac{\partial f(x, y, z)}{\partial x}dx + \frac{\partial f(x, y, z)}{\partial y}dy + \frac{\partial f(x, y, z)}{\partial z}dz\) This is the differential \(dw\) for the function \(w = f(x, y, z)\).