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

Problem 2.1913 · hard

1. \[ \frac{d}{d x} \left(\frac{3 x^{2}}{4} + x - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]
   Differentiate the function with respect to x.✓ Proved
2. \[ = \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} - \frac{d}{d x} \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \left(\frac{d}{d x} \frac{2}{3} + \frac{d}{d x} 2 x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} \]
   sumApply the sum rule to the first part of the product rule.✓ Proved
5. \[ = - \left(\frac{d}{d x} 2 x + \frac{d}{d x} \frac{3 x^{2}}{2}\right) \ln{\left(3 x + 2 \right)} - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} \]
   constant algebraThe derivative of the constant 2/3 is 0. Simplify the expression inside the parentheses.✓ Proved
6. \[ = - \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} \]
   power algebraApply the power rule to the derivatives of the polynomial terms. Simplify the polynomial derivative.✓ Proved
7. \[ = - \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} - \frac{\left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} \]
   logarithmicApply the chain rule to the derivative of the logarithm.✓ Proved
8. \[ = - \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} - \frac{3 \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right)}{3 x + 2} \]
   chainApply the chain rule to the derivative of 3*x + 2.✓ Proved
9. \[ = - \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{3 x^{2}}{4} - \frac{\frac{9 x^{2}}{2} + 6 x + 2}{3 x + 2} \]
   algebra algebraMultiply the constant 3 into the numerator. Distribute the 3 into the polynomial.✓ Proved
10. \[ = \frac{3 x}{2} - \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} + 1 - \frac{\frac{9 x^{2}}{2} + 6 x + 2}{3 x + 2} \]
    power algebra algebra algebra simplifyEvaluate the derivatives of the first two terms. Simplify the first term. Distribute the negative sign. Note that the fraction can be simplified further if needed, but let's check the final form. Final simplification of the expression.✓ Proved