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.

This page provides a detailed step-by-step guide to finding the derivative of e^{x^2 + 1} using the chain rule. Learn how to apply the chain rule and differentiate exponential functions with this calculus problem. Suitable for Grades 11-12.

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