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

Problem 2.1661 · hard

1. \[ \frac{d}{d x} \left(- 3 x + \left(3 x - \frac{3}{4}\right) \ln{\left(4 x - 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d x} \left(3 x - \frac{3}{4}\right) \ln{\left(4 x - 1 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \left(3 x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} \left(3 x - \frac{3}{4}\right) + \frac{d}{d x} \left(- 3 x\right) \]
   productApply the product rule to the second term.✓ Proved
4. \[ = \left(3 x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \left(- \frac{d}{d x} \frac{3}{4} + \frac{d}{d x} 3 x\right) \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   algebraDistribute the derivative over the subtraction.✓ Proved
5. \[ = \left(3 x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} - \ln{\left(4 x - 1 \right)} \frac{d}{d x} \frac{3}{4} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) \]
   algebraExpand the expression.✓ Proved
6. \[ = \left(3 x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} 3 x + \frac{d}{d x} \left(- 3 x\right) \]
   constantThe derivative of the constant 3/4 is zero.✓ Proved
7. \[ = \left(3 x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 1 \right)} + 3 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   constant-multipleDifferentiate 3*x.✓ Proved
8. \[ = \frac{\left(3 x - \frac{3}{4}\right) \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} + 3 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   chainApply the chain rule to the logarithm.✓ Proved
9. \[ = \frac{4 \left(3 x - \frac{3}{4}\right)}{4 x - 1} + 3 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 3 x\right) \]
   derivativeDifferentiate the inner function 4*x - 1.✓ Proved
10. \[ = 3 \ln{\left(4 x - 1 \right)} + \frac{d}{d x} \left(- 3 x\right) + \frac{12 x - 3}{4 x - 1} \]
    algebraMultiply the terms in the last part.✓ Proved
11. \[ = 3 \ln{\left(4 x - 1 \right)} - 3 + \frac{12 x - 3}{4 x - 1} \]
    derivative algebraDifferentiate the first term. Factor out 3 from the numerator.✓ Proved
12. \[ = 3 \ln{\left(4 x - 1 \right)} \]
    simplify simplifyCancel the common factor (4*x - 1). Combine the constant terms.✓ Proved