To find the derivative of the function \( e^{x^2 + y^2} \) with respect to \( x \), we'll need to apply the chain rule.

Given:
\[
f(x, y) = e^{x^2 + y^2}
\]

We are finding the partial derivative with respect to \( x \), assuming \( y \) is a constant.

### Step-by-step solution:
1. **Differentiate the exponential function**:
   \[
   \frac{\partial}{\partial x} e^{x^2 + y^2} = e^{x^2 + y^2} \cdot \frac{\partial}{\partial x}(x^2 + y^2)
   \]
   
2. **Differentiate \( x^2 + y^2 \) with respect to \( x \)**:
   \[
   \frac{\partial}{\partial x}(x^2 + y^2) = 2x
   \]
   (Since the derivative of \( y^2 \) with respect to \( x \) is 0, because \( y \) is constant.)

3. **Combine the results**:
   \[
   \frac{\partial}{\partial x} e^{x^2 + y^2} = e^{x^2 + y^2} \cdot 2x
   \]

Thus, the derivative of \( e^{x^2 + y^2} \) with respect to \( x \) is:
\[
\frac{\partial}{\partial x} e^{x^2 + y^2} = 2x e^{x^2 + y^2}
\]

Would you like further details or clarification?

Here are 5 related questions to deepen your understanding:
1. What is the derivative of \( e^{x^2 - y^2} \) with respect to \( x \)?
2. How would you find the derivative of \( e^{x^2 + y^2} \) with respect to \( y \)?
3. Can you find the second derivative of \( e^{x^2 + y^2} \) with respect to \( x \)?
4. What is the total differential of \( e^{x^2 + y^2} \)?
5. How would you approach finding the gradient of \( e^{x^2 + y^2} \)?

**Tip:** When differentiating exponential functions, always remember to apply the chain rule, especially when the exponent is a function of multiple variables.

Explore the step-by-step solution to find the partial derivative of e^(x^2 + y^2) with respect to x. This calculus problem uses the chain rule to differentiate exponential functions of multiple variables, suitable for undergraduate students studying calculus.

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