Let's break down the problem and solve each part carefully.

Part A: Use the chain rule to find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$

Given:

• $z = x^2 \sin(y)$
• $x = 5s^2 + 5t^2$
• $y = -6st$

Step 1: Use the Chain Rule for Partial Derivatives

To find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$, we will apply the chain rule:

$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$ $\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}$

Step 2: Calculate the Necessary Derivatives

First, find the partial derivatives of $z$ with respect to $x$ and $y$:

$\frac{\partial z}{\partial x} = 2x \sin(y)$ $\frac{\partial z}{\partial y} = x^2 \cos(y)$

Now, find the partial derivatives of $x$ and $y$ with respect to $s$ and $t$:

$\frac{\partial x}{\partial s} = 10s$ $\frac{\partial x}{\partial t} = 10t$ $\frac{\partial y}{\partial s} = -6t$ $\frac{\partial y}{\partial t} = -6s$

Step 3: Apply the Chain Rule

Now, plug these into the chain rule formulas:

$\frac{\partial z}{\partial s} = (2x \sin(y)) \cdot (10s) + (x^2 \cos(y)) \cdot (-6t)$ $\frac{\partial z}{\partial t} = (2x \sin(y)) \cdot (10t) + (x^2 \cos(y)) \cdot (-6s)$

Part B: Find the Numerical Values When $(s,t) = (2, 1)$

Now, to find the numerical values, we need to substitute $s = 2$ and $t = 1$ into the equations for $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

Step 1: Calculate $x$ and $y$ at $(s,t) = (2,1)$

$x = 5(2)^2 + 5(1)^2 = 5(4) + 5(1) = 20 + 5 = 25$ $y = -6(2)(1) = -12$

Step 2: Substitute into the Derivative Formulas

Now, substitute $x = 25$ and $y = -12$ into the formulas for the derivatives.

$\frac{\partial z}{\partial s} = (2 \times 25 \times \sin(-12)) \cdot (10 \times 2) + (25^2 \times \cos(-12)) \cdot (-6 \times 1)$ $\frac{\partial z}{\partial t} = (2 \times 25 \times \sin(-12)) \cdot (10 \times 1) + (25^2 \times \cos(-12)) \cdot (-6 \times 2)$

Now let's calculate these step by step.

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Would you like me to calculate the exact numerical values for the derivatives?