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The problem asks us to find \(\left(\frac{\partial w}{\partial x}\right)_{y,z}\), where \( w = x^2 + y - z + \sin t \) and \( x + y = t \).

### Step-by-step solution:

#### 1. Recognize the problem structure:
- \( w = x^2 + y - z + \sin t \)
- The constraint: \( x + y = t \) (which implicitly defines \( y \) in terms of \( x \) and \( t \)).

#### 2. Apply the constraint:
From \( x + y = t \), we can express \( y \) as:
\[
y = t - x
\]
This allows us to replace \( y \) in the function \( w \) with \( t - x \).

Thus, \( w \) becomes:
\[
w = x^2 + (t - x) - z + \sin t
\]
Simplifying:
\[
w = x^2 - x + t - z + \sin t
\]

#### 3. Differentiate \( w \) with respect to \( x \):
Now, to find \( \frac{\partial w}{\partial x} \), we differentiate the simplified \( w \) expression with respect to \( x \), treating \( t \) and \( z \) as constants.

\[
\frac{\partial w}{\partial x} = 2x - 1
\]

Thus, the answer is:
\[
\boxed{2x - 1}
\]

Let me know if you'd like further details, or if you have more questions!

Here are some related questions to think about:
1. How does the chain rule apply when functions involve constraints like \( x + y = t \)?
2. What would happen if \( t \) depended on \( x \) (i.e., if \( t = t(x) \))?
3. Can you compute \(\frac{\partial w}{\partial y}\) directly from the original expression?
4. How would the problem change if \( w \) had terms like \( x^2y \)?
5. What if \( w \) also depended on \( z \) non-linearly (e.g., \( z^2 \))?

**Tip:** When dealing with constrained functions, always remember to incorporate the constraints before differentiating!

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

Learn how to find the partial derivative (∂w/∂x)_y,z where w = x^2 + y - z + sin t, given the constraint x + y = t. This detailed solution uses key multivariable calculus concepts like partial differentiation and implicit differentiation. Suitable for undergraduate students.

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