Derivative of \( \displaystyle \ln{\left(\frac{e^{3 x + 1}}{3 x + 1} \right)} \)

Problem 2.1454 · hard

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{3 x + 1}}{3 x + 1} \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(3 x - \ln{\left(3 x + 1 \right)} + 1\right) \]
   logarithmic simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(3*x + 1)) to 3*x + 1.✓ Proved
3. \[ = \frac{d}{d x} \left(3 x + 1\right) - \frac{d}{d x} \ln{\left(3 x + 1 \right)} \]
   sumSplit the derivative into two parts.✓ Proved
4. \[ = 3 - \frac{d}{d x} \ln{\left(3 x + 1 \right)} \]
   constantThe derivative of 3*x + 1 is 3.✓ Proved
5. \[ = 3 - \frac{\frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = 3 - \frac{3}{3 x + 1} \]
   derivative algebraThe derivative of 3*x + 1 is 3. Simplify the expression.✓ Proved
7. \[ = \frac{9 x}{3 x + 1} \]
   algebra algebra simplifyFind a common denominator. Distribute the 3. Simplify the numerator.✓ Proved