Use the Chain Rule to find $$ \frac{d z}{d t} $$, where $$ z=3 x \cos y, x=t^{5} $$, and $$ y=6 t^{4} $$. $$ \frac{\partial z}{\partial x}=3 \cos y $$ (Type an expression using $$ x $$ and $$ y $$ as the variables.) $$ \frac{d x}{d t}=5 t^{4} $$ (Type an expression using $$ t $$ as the variable.) $$ \frac{\partial z}{\partial y}=\square $$ (Type an expression using $$ x $$ and $$ y $$ as the variables.)

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To find $$ \frac{d z}{d t} $$ using the Chain Rule, we need to calculate the partial derivatives of $$ z $$ with respect to $$ x $$ and $$ y $$, and then use the Chain Rule formula.

Given:
$$ z = 3x \cos y $$
$$ x = t^5 $$
$$ y = 6t^4 $$

First, let's find the partial derivatives of $$ z $$ with respect to $$ x $$ and $$ y $$:
1. $$ \frac{\partial z}{\partial x} = 3 \cos y $$
2. $$ \frac{\partial z}{\partial y} = -3x \sin y $$

Now, we can use the Chain Rule formula:
$$ \frac{d z}{d t} = \frac{\partial z}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial z}{\partial y} \cdot \frac{d y}{d t} $$

Substitute the given values:
$$ \frac{d z}{d t} = (3 \cos y) \cdot (5t^4) + (-3x \sin y) \cdot (24t^3) $$

Substitute $$ x = t^5 $$ and $$ y = 6t^4 $$ into the expression:
$$ \frac{d z}{d t} = (3 \cos (6t^4)) \cdot (5t^4) + (-3t^5 \sin (6t^4)) \cdot (24t^3) $$

Therefore, the expression for $$ \frac{d z}{d t} $$ is:
$$ \frac{d z}{d t} = 15t^4 \cos (6t^4) - 72t^8 \sin (6t^4) $$
$$ \frac{d z}{d t} = 15t^4 \cos (6t^4) - 72t^8 \sin (6t^4) $$

Use the Chain Rule to find , where , and .

(Type an expression using and as the variables.)

(Type an expression using as the variable.)

(Type an expression using and as the variables.)

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