Use the Chain Rule to find \( \frac{d z}{d t} \), where \( z=3 x \sin y, x=t^{4} \), and \( y=3 t^{2} \). \( \frac{d z}{d t}=\square \) (Type an expression using tas the variable. Use parentheses to clearly denote the argument of each function.)

To find \( \frac{d z}{d t} \) using the Chain Rule, we need to differentiate the function \( z = 3x \sin y \) with respect to \( t \), where \( x = t^4 \) and \( y = 3t^2 \). Given: - \( z = 3x \sin y \) - \( x = t^4 \) - \( y = 3t^2 \) Using the Chain Rule, we can find \( \frac{d z}{d t} \) by differentiating \( z \) with respect to \( t \) and then applying the chain rule to account for the intermediate variables \( x \) and \( y \). Let's start by finding \( \frac{d z}{d t} \) using the Chain Rule. Evaluate the derivative by following steps: - step0: Solution: \(\frac{d}{dt}\left(3t^{4}\sin\left(3t^{2}\right)\right)\) - step1: Use differentiation rules: \(\frac{d}{dt}\left(3\right)\times t^{4}\sin\left(3t^{2}\right)+3\times \frac{d}{dt}\left(t^{4}\right)\times \sin\left(3t^{2}\right)+3t^{4}\times \frac{d}{dt}\left(\sin\left(3t^{2}\right)\right)\) - step2: Find the derivative: \(0\times t^{4}\sin\left(3t^{2}\right)+3\times \frac{d}{dt}\left(t^{4}\right)\times \sin\left(3t^{2}\right)+3t^{4}\times \frac{d}{dt}\left(\sin\left(3t^{2}\right)\right)\) - step3: Calculate: \(0+3\times \frac{d}{dt}\left(t^{4}\right)\times \sin\left(3t^{2}\right)+3t^{4}\times \frac{d}{dt}\left(\sin\left(3t^{2}\right)\right)\) - step4: Calculate: \(0+3\times 4t^{3}\sin\left(3t^{2}\right)+3t^{4}\times \frac{d}{dt}\left(\sin\left(3t^{2}\right)\right)\) - step5: Calculate: \(0+12t^{3}\sin\left(3t^{2}\right)+3t^{4}\times \frac{d}{dt}\left(\sin\left(3t^{2}\right)\right)\) - step6: Calculate: \(0+12t^{3}\sin\left(3t^{2}\right)+3t^{4}\times 6t\cos\left(3t^{2}\right)\) - step7: Calculate: \(0+12t^{3}\sin\left(3t^{2}\right)+18t^{5}\cos\left(3t^{2}\right)\) - step8: Remove 0: \(12t^{3}\sin\left(3t^{2}\right)+18t^{5}\cos\left(3t^{2}\right)\) The derivative of \( z \) with respect to \( t \) using the Chain Rule is \( \frac{d z}{d t} = 12t^{3}\sin(3t^{2}) + 18t^{5}\cos(3t^{2}) \).