Find the derivative \( \frac{d w}{d t} \), where \( w=6 x y z, x=4 t^{4}, y=t^{-1} \), and \( z=2 t^{-3} \). \( \frac{\partial w}{\partial x}=6 y z \) (Type an expression using \( x, y \), and \( z \) as the variables.) \( \frac{\partial w}{\partial y}= \) (Type an expression using \( x, y \), and \( z \) as the variables.)

To find the derivative \( \frac{d w}{d t} \), we'll use the chain rule. First, let's express \( w \) in terms of \( t \): Given \( w = 6xy z \), and substituting \( x = 4t^4 \), \( y = t^{-1} \), and \( z = 2t^{-3} \), we have: \[ w = 6 (4t^4)(t^{-1})(2t^{-3}) = 6 \cdot 4 \cdot 2 \cdot t^{4 - 1 - 3} = 48 t^0 = 48. \] Since \( w \) equals a constant, its derivative with respect to \( t \) is: \[ \frac{d w}{d t} = 0. \] Next, let's compute \( \frac{\partial w}{\partial y} \): Given that \( \frac{\partial w}{\partial x} = 6yz \), we can find \( \frac{\partial w}{\partial y} \) directly from the equation for \( w \): \[ \frac{\partial w}{\partial y} = 6xy z. \] So, the expression for \( \frac{\partial w}{\partial y} \) using \( x, y, \) and \( z \) is: \[ \frac{\partial w}{\partial y} = 6xyz. \]