Use the Chain Rule to find the indicated partial derivatives.
P = 
	u2 + v2 + w2
,    u = xey,    v = yex,    w = exy;
 
∂P
∂x
, 
∂P
∂y
    when x = 0, y = 3
∂P
∂x
	 = 	
∂P
∂y
	 = 	


We are tasked with finding the partial derivatives of the function \( P = u^2 + v^2 + w^2 \), where \( u = x e^y \), \( v = y e^x \), and \( w = e^{xy} \), using the Chain Rule.

### Step 1: Compute the partial derivative of \( P \) with respect to \( x \)

The Chain Rule tells us that:

\[
\frac{\partial P}{\partial x} = 2u \frac{\partial u}{\partial x} + 2v \frac{\partial v}{\partial x} + 2w \frac{\partial w}{\partial x}
\]

#### 1.1: Derivatives of \( u \), \( v \), and \( w \) with respect to \( x \)

- \( u = x e^y \)
 

Use the Chain Rule to find the indicated partial derivatives. P = u^2 + v^2 + w^2, u = xey, v = yex, w = exy; ∂P/∂x, ∂P/∂y when x = 0, y = 3. ∂P/∂x = ∂P/∂y = 

Learn how to apply the Chain Rule to find the partial derivatives of P = u^2 + v^2 + w^2, where u = x e^y, v = y e^x, and w = e^{xy}. This step-by-step solution covers multivariable calculus concepts, including partial derivatives and the Chain Rule. Example given for x = 0, y = 3.

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