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

Problem 2.205 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(4 x - 3 \right)} - x - \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} - 1 \]
   constantThe derivative of x is 1.✓ Proved
4. \[ = \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
   constant-multiplePull out the constant factor 3/4.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} x - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
   derivative algebraThe derivative of x is 1. Rearrange the terms.✓ Proved
7. \[ = \left(x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} - 1 \]
   algebraGroup the terms containing the derivative.✓ Proved
8. \[ = \frac{\left(x - \frac{3}{4}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} + \ln{\left(4 x - 3 \right)} - 1 \]
   chainApply the chain rule to the logarithm.✓ Proved
9. \[ = \ln{\left(4 x - 3 \right)} \]
   derivative algebra algebra simplifyThe derivative of 4*x - 3 is 4. Distribute the 4 in the numerator. Simplify the fraction. Final simplification.✓ Proved