Find $$ \partial w / \partial s $$ and $$ \partial w / \partial t $$ using the appropriate Chain Rule. $$ \begin{array}{l}\begin{array}{l}\text { Function } \ w=\sin (4 x+5 y) \ x=s+t, y=s-t\end{array} \ \begin{array}{l}\frac{\partial w}{\partial s}= \ \frac{\partial w}{\partial t}=\end{array} \ \text { Evaluate each partial derivative at the given values of } s \text { and } t \text {. } \ \frac{\partial w}{\partial s}=\square \ \frac{\partial w}{\partial t}=\end{array} $$

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We start with the function

$$
w = \sin(4x + 5y)
$$

with

$$
x = s + t \quad \text{and} \quad y = s - t.
$$

**Step 1. Compute the necessary derivatives of $$x$$ and $$y$$:**

$$
\frac{\partial x}{\partial s} = 1, \quad \frac{\partial y}{\partial s} = 1,
$$
$$
\frac{\partial x}{\partial t} = 1, \quad \frac{\partial y}{\partial t} = -1.
$$

**Step 2. Write the chain rule formula for $$\frac{\partial w}{\partial s}$$ and $$\frac{\partial w}{\partial t}$$:**

Using the chain rule,

$$
\frac{\partial w}{\partial s} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial s},
$$
$$
\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial t}.
$$

**Step 3. Compute $$\frac{\partial w}{\partial x}$$ and $$\frac{\partial w}{\partial y}$$:**

Since

$$
w = \sin(4x + 5y),
$$

by the chain rule we have

$$
\frac{\partial w}{\partial x} = \cos(4x+5y) \cdot 4,
$$
$$
\frac{\partial w}{\partial y} = \cos(4x+5y) \cdot 5.
$$

**Step 4. Substitute into the chain rule formulas:**

For $$\frac{\partial w}{\partial s}$$:

$$
\frac{\partial w}{\partial s} = \left[\cos(4x+5y) \cdot 4\right](1) + \left[\cos(4x+5y) \cdot 5\right](1).
$$

Combine the terms:

$$
\frac{\partial w}{\partial s} = \cos(4x+5y)(4 + 5) = 9\cos(4x+5y).
$$

Now, recall that $$ x = s + t $$ and $$ y = s - t $$, so

$$
\frac{\partial w}{\partial s} = 9\cos\big(4(s+t)+5(s-t)\big).
$$

For $$\frac{\partial w}{\partial t}$$:

$$
\frac{\partial w}{\partial t} = \left[\cos(4x+5y) \cdot 4\right](1) + \left[\cos(4x+5y) \cdot 5\right](-1).
$$

Combine the terms:

$$
\frac{\partial w}{\partial t} = \cos(4x+5y)(4 - 5) = -\cos(4x+5y).
$$

Substitute back $$ x = s+t $$ and $$ y = s-t $$:

$$
\frac{\partial w}{\partial t} = -\cos\big(4(s+t)+5(s-t)\big).
$$

**Final Answers:**

$$
\frac{\partial w}{\partial s} = 9\cos\big(4(s+t)+5(s-t)\big),
$$
$$
\frac{\partial w}{\partial t} = -\cos\big(4(s+t)+5(s-t)\big).
$$
$$ \frac{\partial w}{\partial s} = 9\cos(9s - t), $$ $$ \frac{\partial w}{\partial t} = -\cos(9s - t). $$

Find and using the appropriate Chain Rule.

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