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Given:
\[
w = x^2 + y - z + \sin t
\]
and the relationship \(x + y = t\), we need to find the partial derivative \(\left( \frac{\partial w}{\partial x} \right)_{y, z}\).

### Step-by-Step Solution

1. **Express \(t\) in terms of \(x\) and \(y\):**
   \[
   t = x + y.
   \]
   
2. **Substitute \(t\) into the equation for \(w\):**
   Since \(t = x + y\), we have:
   \[
   w = x^2 + y - z + \sin(x + y).
   \]

3. **Differentiate \(w\) with respect to \(x\):**
   Since we are finding \(\frac{\partial w}{\partial x}\) while keeping \(y\) and \(z\) constant, we treat \(y\) and \(z\) as constants during differentiation:
   \[
   \frac{\partial w}{\partial x} = \frac{\partial}{\partial x} \left( x^2 + y - z + \sin(x + y) \right).
   \]
   
4. **Apply the derivative:**
   - The derivative of \(x^2\) with respect to \(x\) is \(2x\).
   - The derivative of \(y\) (a constant with respect to \(x\)) is 0.
   - The derivative of \(-z\) (a constant with respect to \(x\)) is 0.
   - The derivative of \(\sin(x + y)\) with respect to \(x\) is \(\cos(x + y) \cdot \frac{\partial}{\partial x} (x + y) = \cos(x + y)\), since the derivative of \(x + y\) with respect to \(x\) is 1.

   Combining these results:
   \[
   \frac{\partial w}{\partial x} = 2x + \cos(x + y).
   \]

### Final Answer
\[
\left( \frac{\partial w}{\partial x} \right)_{y, z} = 2x + \cos(x + y).
\]

Would you like further details or have any questions?

### Related Questions
1. How would the derivative change if \(w\) was a function of \(x\), \(y\), and \(t\) directly?
2. What is the partial derivative of \(w\) with respect to \(y\) given the same conditions?
3. How would you find \(\left(\frac{\partial w}{\partial t}\right)_{x, y, z}\)?
4. If \(z\) also depended on \(x\) and \(y\), how would that affect the partial derivative \(\left(\frac{\partial w}{\partial x}\right)_{y, z}\)?
5. Can we interpret this result geometrically in terms of changes in \(w\) as \(x\) varies?

### Tip
Always keep track of which variables are held constant when taking partial derivatives; it helps in understanding how the function changes with respect to one specific variable.

Find (∂w/∂x)_{y,z} if w = x^2 + y - z + sin(t) and x + y = t.

This problem involves calculating the partial derivative of w with respect to x, where w is a function of x, y, z, and t. The solution applies the chain rule and simplifies the derivative, resulting in 2x + cos(x + y). Suitable for college-level calculus students.

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