suppose Z=f(x,y,u(x,y),v(x,y)) = x^2y + u(x,y)*sin(x) + v(x,y)*e^(x^2) + 2u(x,y). calculate the partial derivative of Z with respect to x

To calculate the partial derivative of \( Z \) with respect to \( x \), we need to differentiate the given function \( Z = f(x, y, u(x,y), v(x,y)) \) with respect to \( x \) while treating \( y \) as a constant. Additionally, we must consider the dependence of \( u(x,y) \) and \( v(x,y) \) on \( x \) and \( y \).

The function is given by:
\[
Z = x^2y + u(x,y)\sin(x) + v(x,y)e^{x^2} + 2u(x,y)
\]

### Step 1: Differentiate each term with respect to \( x \).

1. **First term: \( x^2y \)**
   \[
   \frac{\partial}{\partial x}(x^2y) = 2xy
   \]

2. **Second term: \( u(x,y)\sin(x) \)**
   Apply the product rule: \( \frac{d}{dx}(f \cdot g) = f' \cdot g + f \cdot g' \)
   \[
   \frac{\partial}{\partial x}\left(u(x,y)\sin(x)\right) = \frac{\partial u(x,y)}{\partial x}\sin(x) + u(x,y)\cos(x)
   \]

3. **Third term: \( v(x,y)e^{x^2} \)**
   Again, apply the product rule:
   \[
   \frac{\partial}{\partial x}\left(v(x,y)e^{x^2}\right) = \frac{\partial v(x,y)}{\partial x}e^{x^2} + v(x,y)\cdot 2xe^{x^2}
   \]

4. **Fourth term: \( 2u(x,y) \)**
   \[
   \frac{\partial}{\partial x}\left(2u(x,y)\right) = 2\frac{\partial u(x,y)}{\partial x}
   \]

### Step 2: Combine all the results
Now, sum the partial derivatives of each term:
\[
\frac{\partial Z}{\partial x} = 2xy + \frac{\partial u(x,y)}{\partial x}\sin(x) + u(x,y)\cos(x) + \frac{\partial v(x,y)}{\partial x}e^{x^2} + v(x,y)\cdot 2xe^{x^2} + 2\frac{\partial u(x,y)}{\partial x}
\]

### Step 3: Simplify the expression
Combine like terms involving \(\frac{\partial u(x,y)}{\partial x}\):
\[
\frac{\partial Z}{\partial x} = 2xy + \left(\sin(x) + 2\right)\frac{\partial u(x,y)}{\partial x} + u(x,y)\cos(x) + \frac{\partial v(x,y)}{\partial x}e^{x^2} + 2xv(x,y)e^{x^2}
\]

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

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Would you like to explore any part of the solution in more detail, or do you have any questions?

Here are 5 related questions you might consider:
1. How would you find the partial derivative of \( Z \) with respect to \( y \)?
2. What happens if \( u(x, y) \) and \( v(x, y) \) are given specific forms, like \( u(x, y) = x + y \)?
3. How would the result change if \( Z \) were a function of more variables?
4. Can you generalize this approach for functions involving more complex dependencies?
5. How does the chain rule apply when both \( u(x,y) \) and \( v(x,y) \) are implicit functions?

**Tip:** When differentiating functions of multiple variables, always keep track of which variables are held constant and which are not.

Learn how to calculate the partial derivative of Z with respect to x step-by-step. This problem involves partial derivatives, the product rule, and exponential functions. Suitable for advanced undergraduate students and beyond.

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