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.