Derivative of \( \displaystyle - 3 x^{2} + \frac{3 x}{2} + \left(6 x^{2} - 3 x + \frac{3}{8}\right) \ln{\left(4 x - 1 \right)} \)

Problem 2.2019 · hard

1. \[ \frac{d}{d x} \left(- 3 x^{2} + \frac{3 x}{2} + \left(6 x^{2} - 3 x + \frac{3}{8}\right) \ln{\left(4 x - 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{d}{d x} \left(6 x^{2} - 3 x + \frac{3}{8}\right) \ln{\left(4 x - 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(6 x^{2} - 3 x + \frac{3}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(6 x^{2} - 3 x + \frac{3}{8}\right) + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]
   productApply the product rule to the third term.✓ Proved
4. \[ = \left(6 x^{2} - 3 x + \frac{3}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \left(\frac{d}{d x} \frac{3}{8} - \frac{d}{d x} 3 x + \frac{d}{d x} 6 x^{2}\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]
   sumApply the sum rule to the polynomial derivative.✓ Proved
5. \[ = \left(- \frac{d}{d x} 3 x + \frac{d}{d x} 6 x^{2}\right) \ln{\left(4 x - 1 \right)} + \left(6 x^{2} - 3 x + \frac{3}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]
   constant algebraThe derivative of the constant 3/8 is zero. Simplify the expression inside the parenthesis.✓ Proved
6. \[ = \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \left(6 x^{2} - 3 x + \frac{3}{8}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) \]
   powerDifferentiate the polynomial terms.✓ Proved
7. \[ = \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{\left(6 x^{2} - 3 x + \frac{3}{8}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- 3 x^{2}\right) + \frac{4 \left(6 x^{2} - 3 x + \frac{3}{8}\right)}{4 x - 1} \]
   derivativeDifferentiate the inner function 4*x - 1.✓ Proved
9. \[ = - 6 x + \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{3}{2} + \frac{24 x^{2} - 12 x + \frac{3}{2}}{4 x - 1} \]
   derivativeEvaluate the derivatives of the first two terms and distribute the 4.✓ Proved
10. \[ = - 6 x + \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{3}{2} + \frac{48 x^{2} - 24 x + 3}{8 x - 2} \]
    algebra algebraMultiply the numerator and denominator by 2 to clear the fraction in the last term. Distribute the 2 in the denominator.✓ Proved
11. \[ = - 6 x + \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{3}{2} + \frac{24 x^{2} - 12 x + \frac{3}{2}}{4 x - 1} \]
    algebraRewrite the last term for easier simplification.✓ Proved
12. \[ = - 6 x + \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{3}{2} + \frac{48 x^{2} - 24 x + 3}{8 x - 2} \]
    algebra algebraConvert back to a common denominator form. Expand the denominator.✓ Proved
13. \[ = - 6 x + \left(12 x - 3\right) \ln{\left(4 x - 1 \right)} + \frac{3}{2} + \frac{24 x^{2} - 12 x + \frac{3}{2}}{4 x - 1} \]
    simplifyFinal simplified form.✓ Proved