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

Problem 2.1925 · hard

1. \[ \frac{d}{d x} \left(2 x \ln{\left(4 x + 2 \right)} - 2 x + \ln{\left(2 x + 1 \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = \frac{d}{d x} 2 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
   constantDifferentiate the linear term -2*x.✓ Proved
4. \[ = 2 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 2 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + 2 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
   constant-multipleDifferentiate the term 2*x.✓ Proved
6. \[ = \frac{2 x \frac{d}{d x} \left(4 x + 2\right)}{4 x + 2} + 2 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = \frac{8 x}{4 x + 2} + 2 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \ln{\left(2 x + 1 \right)} - 2 \]
   derivativeDifferentiate the inner function 4*x + 2.✓ Proved
8. \[ = \frac{8 x}{4 x + 2} + 2 \ln{\left(4 x + 2 \right)} - 2 + \frac{\frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} \]
   chainApply the chain rule to the second logarithm.✓ Proved
9. \[ = \frac{8 x}{4 x + 2} + 2 \ln{\left(4 x + 2 \right)} - 2 + \frac{2}{2 x + 1} \]
   derivative algebraDifferentiate the inner function 2*x + 1. Multiply the terms in the numerator.✓ Proved
10. \[ = \frac{4 x}{2 x + 1} + 2 \ln{\left(4 x + 2 \right)} - 2 + \frac{2}{2 x + 1} \]
    algebraSimplify the fraction (8*x)/(4*x + 2) by dividing by 2.✓ Proved
11. \[ = 2 \ln{\left(4 x + 2 \right)} - 2 + \frac{4 x + 2}{2 x + 1} \]
    algebra algebraCombine the fractions with the same denominator. Factor the numerator 4*x + 2.✓ Proved
12. \[ = 2 \ln{\left(4 x + 2 \right)} \]
    algebra simplifyCancel the common factor (2*x + 1). Final simplification.✓ Proved