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

Problem 2.429 · hard

1. \[ \frac{d}{d x} \ln{\left(\left(2 x + 2\right)^{2} + 1 \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\left(2 x + 2\right)^{2} + 1\right)}{\left(2 x + 2\right)^{2} + 1} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x + 2\right)^{2}}{\left(2 x + 2\right)^{2} + 1} \]
   sumApply the sum rule to the inner expression.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(2 x + 2\right)^{2}}{\left(2 x + 2\right)^{2} + 1} \]
   constantThe derivative of a constant is zero.✓ Proved
5. \[ = \frac{\left(4 x + 4\right) \frac{d}{d x} \left(2 x + 2\right)}{\left(2 x + 2\right)^{2} + 1} \]
   powerApply the power rule to the squared term.✓ Proved
6. \[ = \frac{8 x + 8}{\left(2 x + 2\right)^{2} + 1} \]
   derivative algebra algebraDifferentiate the linear term inside the parenthesis. Multiply the terms together. Distribute the 4 in the numerator.✓ Proved
7. \[ = \frac{8 x + 8}{4 x^{2} + 8 x + 5} \]
   simplifyExpand the denominator and simplify.✓ Proved