\( 1 \leftarrow \) Use the Chain Rule to find \( \frac{d z}{d t} \), where \( z=3 x \sin y, x=t^{4} \), and \( y=3 t^{2} \). Save

To find \( \frac{d z}{d t} \) using the Chain Rule, start by determining the partial derivatives of \( z \) with respect to \( x \) and \( y \): 1. \( \frac{\partial z}{\partial x} = 3 \sin y \) 2. \( \frac{\partial z}{\partial y} = 3 x \cos y \) Next, compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \): 1. \( x = t^4 \) so \( \frac{dx}{dt} = 4t^3 \) 2. \( y = 3t^2 \) so \( \frac{dy}{dt} = 6t \) Now apply the Chain Rule: \[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt} \] Substituting the values: \[ \frac{dz}{dt} = (3 \sin y)(4t^3) + (3 x \cos y)(6t) \] Now replace \( x \) and \( y \) with their expressions in terms of \( t \): \[ = 12t^3 \sin(3t^2) + 18t^4 \cos(3t^2) \] Thus, \( \frac{dz}{dt} = 12t^3 \sin(3t^2) + 18t^4 \cos(3t^2) \).