VLM 15.4 The Chain Question 3, 15.4.11-Setup \& Solve HW Score: 1.82\%, 0.4 of 22 Part 1 of 5 points Points: 0 of 1 Use the Chain Rule to find \( \frac{d w}{d t} \), where \( w=\sin (12 x) \cos (2 y), x=\frac{t}{4} \), and \( y=t^{5} \). \[ \frac{\partial w}{\partial x}=\square \] (Type an expression using \( x \) and \( y \) as the variables.)

To solve for \(\frac{\partial w}{\partial x}\), we first use the chain rule. Given \(w = \sin(12x) \cos(2y)\), we need to find the partial derivative of \(w\) with respect to \(x\): \[ \frac{\partial w}{\partial x} = 12 \cos(12x) \cos(2y). \] This is because when differentiating \(\sin(12x)\), we multiply by the derivative of \(12x\), which is 12. Next, since we are looking for \(\frac{d w}{d t}\), we'll need to find \(\frac{\partial w}{\partial y}\) later, but for now, we can box our answer for \(\frac{\partial w}{\partial x}\): \[ \frac{\partial w}{\partial x} = 12 \cos(12x) \cos(2y). \]