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The problem is asking for \(\left( \frac{\partial w}{\partial x} \right)_{y,z}\) given that:

\[
w = x^2 + y - z + \sin t
\]
and 
\[
x + y = t.
\]

Here’s how we approach it:

### Step 1: Implicit Differentiation
The equation \( x + y = t \) implicitly defines \( y \) as a function of \( x \) and \( t \). Therefore, when differentiating \( w \) with respect to \( x \), we need to account for the dependency

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

Solve the problem to find (∂w/∂x)_y,z for w = x^2 + y - z + sin(t) and the constraint x + y = t. Learn how implicit differentiation and partial derivatives are applied to this multivariable calculus problem. Suitable for undergraduate students.

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