Derivative of \( \displaystyle \ln{\left(e^{x + 1} + 1 \right)} \)

Problem 2.435 · hard

1. \[ \frac{d}{d x} \ln{\left(e^{x + 1} + 1 \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(e^{x + 1} + 1\right)}{e^{x + 1} + 1} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} e^{x + 1}}{e^{x + 1} + 1} \]
   sumApply the sum rule to the inner expression.✓ Proved
4. \[ = \frac{\frac{d}{d x} e^{x + 1}}{e^{x + 1} + 1} \]
   constant algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
5. \[ = \frac{e^{x + 1}}{e^{x + 1} + 1} \]
   exponential algebraDifferentiate the exponential function. Combine the terms into a single fraction.✓ Proved