Derivative of \( \displaystyle \ln{\left(e^{5 x + 2} + 1 \right)} \)

Problem 2.618 · hard

1. \[ \frac{d}{d x} \ln{\left(e^{5 x + 2} + 1 \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(e^{5 x + 2} + 1\right)}{e^{5 x + 2} + 1} \]
   chainApply the chain rule for the natural logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} e^{5 x + 2}}{e^{5 x + 2} + 1} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = \frac{\frac{d}{d x} e^{5 x + 2}}{e^{5 x + 2} + 1} \]
   constant algebraThe derivative of the constant 1 is 0. Simplify the expression.✓ Proved
5. \[ = \frac{e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right)}{e^{5 x + 2} + 1} \]
   exponentialApply the chain rule for the exponential function.✓ Proved
6. \[ = \frac{5 e^{5 x + 2}}{e^{5 x + 2} + 1} \]
   derivative algebraDifferentiate the linear function 5*x + 2. Simplify the final expression.✓ Proved