Derivative of \( \displaystyle e^{x^{2} + 1} \)

Problem 2.488 · hard

1. \[ \frac{d}{d x} e^{x^{2} + 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = e^{x^{2} + 1} \frac{d}{d x} \left(x^{2} + 1\right) \]
   chainApply the chain rule.✓ Proved
3. \[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} x^{2}\right) e^{x^{2} + 1} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = \left(2 x \frac{d}{d x} x + \frac{d}{d x} 1\right) e^{x^{2} + 1} \]
   powerApply the power rule to x**2.✓ Proved
5. \[ = 2 x e^{x^{2} + 1} \]
   derivative simplifyDifferentiate x and the constant 1. Simplify the expression.✓ Proved