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.