To find the derivative of the function $e^{x^2 + 1}$, we will use the chain rule.

Step-by-step solution:

Let: $f(x) = e^{x^2 + 1}$

1. The derivative of $e^u$ with respect to $u$ is $e^u$, so we start by treating $x^2 + 1$ as $u$. Thus, the derivative of $e^u$ with respect to $u = x^2 + 1$ is: $\frac{d}{du}(e^u) = e^u$
2. Now, apply the chain rule by taking the derivative of $u = x^2 + 1$ with respect to $x$, which is: $\frac{du}{dx} = \frac{d}{dx}(x^2 + 1) = 2x$
3. Putting it all together using the chain rule: $\frac{d}{dx}\left( e^{x^2 + 1} \right) = e^{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1) = e^{x^2 + 1} \cdot 2x$

Final result:

$\frac{d}{dx}\left( e^{x^2 + 1} \right) = 2x e^{x^2 + 1}$

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. Find the derivative of $e^{3x^2 + 2x}$.
2. Find the derivative of $e^{x^3 + x}$.
3. Differentiate $e^{\sin x}$.
4. Compute the derivative of $e^{x^2 + x + 1}$.
5. Find the second derivative of $e^{x^2 + 1}$.

Tip: When using the chain rule, always identify the "inside function" and "outside function" for a more systematic approach to differentiation.