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!