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

Let's break it down step by step.

### Step 1: Implicit Relation between \(x\), \(y\), and \(t\)
We are given the relation \(x + y = t\). This can be rewritten as:
\[
y = t - x
\]
This means that \(y\) is not independent of \(x\) but instead depends on \(x\) and \(t\).

### Step 2: Expression for \(w\)
The expression for \(w\) is:
\[
w = x^2 + y - z + \sin t
\]
Now, substitute \(y = t - x\) into this equation:
\[
w = x^2 + (t - x) - z + \sin t
\]
Simplifying:
\[
w = x^2 - x + t - z + \sin t
\]

### Step 3: Partial Derivative of \(w\) with respect to \(x\)
Now, we compute \(\frac{\partial w}{\partial x}\), treating \(z\) and \(t\) as constants. Differentiating term by term:
\[
\frac{\partial}{\partial x}(x^2) = 2x
\]
\[
\frac{\partial}{\partial x}(-x) = -1
\]
\[
\frac{\partial}{\partial x}(t - z + \sin t) = 0 \quad \text{(since \(t\) and \(z\) are independent of \(x\))}
\]

Thus, the partial derivative is:
\[
\frac{\partial w}{\partial x} = 2x - 1
\]

### Final Answer:
\[
\frac{\partial w}{\partial x} = 2x - 1
\]

Would you like a more detailed breakdown or have any further questions?

Here are some related questions:
1. How does the chain rule apply when functions depend on multiple variables?
2. What is the significance of implicit differentiation in this context?
3. How would the result change if \(z\) depended on \(x\) as well?
4. How does \(\sin t\) behave in the context of partial differentiation with respect to \(x\)?
5. What are the geometric interpretations of partial derivatives?

**Tip:** When dealing with implicit relations, always substitute the dependent variables before differentiating to avoid mistakes.

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 when w = x^2 + y - z + sin(t) and x + y = t. This problem applies implicit differentiation and the chain rule to handle the relation between x and y.

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