Derivative of \( \displaystyle \ln{\left(\left(5 x - 3\right)^{2} + 1 \right)} \)

Problem 2.587 · hard

1. \[ \frac{d}{d x} \ln{\left(\left(5 x - 3\right)^{2} + 1 \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(\left(5 x - 3\right)^{2} + 1\right)}{\left(5 x - 3\right)^{2} + 1} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x - 3\right)^{2}}{\left(5 x - 3\right)^{2} + 1} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(5 x - 3\right)^{2}}{\left(5 x - 3\right)^{2} + 1} \]
   constantThe derivative of a constant is zero.✓ Proved
5. \[ = \frac{\left(10 x - 6\right) \frac{d}{d x} \left(5 x - 3\right)}{\left(5 x - 3\right)^{2} + 1} \]
   powerApply the power rule to the squared term.✓ Proved
6. \[ = \frac{50 x - 30}{\left(5 x - 3\right)^{2} + 1} \]
   derivative algebraDifferentiate the linear term inside the parentheses. Multiply the terms together.✓ Proved
7. \[ = \frac{50 x - 30}{25 x^{2} - 30 x + 10} \]
   simplifyExpand and simplify the expression.✓ Proved