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

Problem 2.6 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \ln{\left(4 x + 2 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
   derivativeDifferentiate the term x.✓ Proved
4. \[ = x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
   chain algebraApply the chain rule to log(4*x + 2). Multiply the numerator.✓ Proved
6. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} + \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 1 \]
   constant-multiplePull out the constant 1/2.✓ Proved
7. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} - 1 + \frac{1}{2 x + 1} \]
   chain algebraApply the chain rule to log(2*x + 1). Simplify the constant multiplication.✓ Proved
8. \[ = \frac{2 x}{2 x + 1} + \ln{\left(4 x + 2 \right)} - 1 + \frac{1}{2 x + 1} \]
   algebraSimplify the fraction (4*x)/(4*x + 2) by dividing by 2.✓ Proved
9. \[ = \ln{\left(4 x + 2 \right)} \]
   algebra simplify simplifyCombine the fractions with the same denominator. Simplify the fraction (2*x + 1)/(2*x + 1). Combine the constant terms.✓ Proved