Derivative of \( \displaystyle - \frac{3 x^{2}}{2} - 2 x + \left(3 x^{2} + 4 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \)

Problem 2.1512 · hard

1. \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{2} - 2 x + \left(3 x^{2} + 4 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{d}{d x} \left(3 x^{2} + 4 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \left(3 x^{2} + 4 x + \frac{4}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x^{2} + 4 x + \frac{4}{3}\right) - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(6 x + 4\right) \ln{\left(3 x + 2 \right)} + \left(3 x^{2} + 4 x + \frac{4}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) \]
   derivativeDifferentiate the polynomial part of the product.✓ Proved
5. \[ = \left(6 x + 4\right) \ln{\left(3 x + 2 \right)} - \frac{d}{d x} 2 x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{3 \left(3 x^{2} + 4 x + \frac{4}{3}\right)}{3 x + 2} \]
   chain algebraApply the chain rule to the logarithm. Simplify the derivative of the log term.✓ Proved
6. \[ = - 3 x + \left(6 x + 4\right) \ln{\left(3 x + 2 \right)} - 2 + \frac{3 \left(3 x^{2} + 4 x + \frac{4}{3}\right)}{3 x + 2} \]
   derivativeDifferentiate the remaining power terms.✓ Proved
7. \[ = - 3 x + \left(6 x + 4\right) \ln{\left(3 x + 2 \right)} - 2 + \frac{9 x^{2} + 12 x + 4}{3 x + 2} \]
   algebraDistribute the 3 in the last term.✓ Proved
8. \[ = \left(6 x + 4\right) \ln{\left(3 x + 2 \right)} \]
   algebra simplify simplifyRecognize the numerator as a perfect square. Simplify the fraction. Combine all remaining terms.✓ Proved