\( 1 \leftarrow \quad \) 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.) \( \frac{\partial w}{\partial y}=6 x z \) (Type an expression using \( x, y \), and \( z \) as the variables.) \( \frac{\partial w}{\partial z}=\square \) (Type an expression using \( x, y \), and \( z \) as the variables.)

To find the derivative \( \frac{d w}{d t} \), we can use the chain rule and the product rule since \( w \) depends on \( t \) through \( x, y, \) and \( z \). First, we can find \( \frac{\partial w}{\partial z} \): 1. Given \( w = 6xyz \): - The partial derivative with respect to \( z \) is: \[ \frac{\partial w}{\partial z} = 6xy \] Now, you have the expressions you need: - \( \frac{\partial w}{\partial x} = 6yz \) - \( \frac{\partial w}{\partial y} = 6xz \) - \( \frac{\partial w}{\partial z} = 6xy \)