Partial derivative of Z=f(x,y,u(x,y),v(x,y))=xy + u(x,y)sinx + v(x,y).e^(x^2) with respect to x

To find the partial derivative of \( Z = f(x, y, u(x, y), v(x, y)) = xy + u(x, y)\sin{x} + v(x, y) e^{x^2} \) with respect to \( x \), we'll apply the chain rule and differentiate each term in \( Z \) with respect to \( x \).

### Step-by-step differentiation:

1. **For the term \( xy \):**
   \[
   \frac{\partial}{\partial x} (xy) = y
   \]

2. **For the term \( u(x, y)\sin{x} \):**
   Using the product rule, we treat \( u(x, y) \) as a function of \( x \) and \( y \), so:
   \[
   \frac{\partial}{\partial x} \left(u(x, y)\sin{x}\right) = \frac{\partial u}{\partial x} \sin{x} + u(x, y) \cos{x}
   \]

3. **For the term \( v(x, y)e^{x^2} \):**
   Again, using the product rule:
   \[
   \frac{\partial}{\partial x} \left(v(x, y)e^{x^2}\right) = \frac{\partial v}{\partial x} e^{x^2} + v(x, y) \cdot \frac{d}{dx} \left(e^{x^2}\right)
   \]
   The derivative of \( e^{x^2} \) with respect to \( x \) is:
   \[
   \frac{d}{dx} \left(e^{x^2}\right) = 2x e^{x^2}
   \]
   So, the derivative of \( v(x, y)e^{x^2} \) is:
   \[
   \frac{\partial v}{\partial x} e^{x^2} + v(x, y) \cdot 2x e^{x^2}
   \]

### Combining all terms:

Now, summing up all the partial derivatives, we get:
\[
\frac{\partial Z}{\partial x} = y + \frac{\partial u}{\partial x} \sin{x} + u(x, y)\cos{x} + \frac{\partial v}{\partial x} e^{x^2} + 2x v(x, y) e^{x^2}
\]

This is the partial derivative of \( Z \) with respect to \( x \).

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Do you need further clarification or more details on any step?

Here are some related questions to consider:
1. What is the partial derivative of \( Z \) with respect to \( y \)?
2. How would you compute the mixed partial derivative \( \frac{\partial^2 Z}{\partial x \partial y} \)?
3. How does the chain rule apply to functions with nested dependencies like \( u(x, y) \) and \( v(x, y) \)?
4. What happens if \( u(x, y) \) and \( v(x, y) \) are given specific functional forms?
5. How would the solution change if we introduced higher-order terms in \( x \)?

**Tip:** Always use the product and chain rules when differentiating products of functions of multiple variables.

Find the partial derivative of Z=f(x, y, u(x, y), v(x, y))=xy + u(x, y)sin(x) + v(x, y)e^(x^2) with respect to x using the chain rule and product rule. This step-by-step solution is suitable for multivariable calculus students and covers essential concepts such as partial derivatives, chain rule, and product rule.

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