Derivative of \( \displaystyle 5 x \ln{\left(4 x + 2 \right)} - 5 x + \frac{5 \ln{\left(2 x + 1 \right)}}{2} \)

Problem 2.2186 · hard

1. \[ \frac{d}{d x} \left(5 x \ln{\left(4 x + 2 \right)} - 5 x + \frac{5 \ln{\left(2 x + 1 \right)}}{2}\right) \]
   sumStart with the derivative of the entire function.✓ Proved
2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \frac{d}{d x} 5 x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constantThe derivative of 5*x is 5.✓ Proved
4. \[ = 5 \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constant-multipleFactor out the constant 5 from the first term.✓ Proved
5. \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + 5 \ln{\left(4 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + 5 \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   derivativeThe derivative of x is 1.✓ Proved
7. \[ = 5 x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + 5 \ln{\left(4 x + 2 \right)} + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constant-multipleFactor out 5/2 from the third term.✓ Proved
8. \[ = \frac{20 x}{4 x + 2} + 5 \ln{\left(4 x + 2 \right)} - 5 + \frac{5}{2 x + 1} \]
   chain algebra algebraApply the chain rule to the logarithmic terms. Simplify the fractions. Distribute the 5 and simplify the terms.✓ Proved
9. \[ = \frac{10 x}{2 x + 1} + 5 \ln{\left(4 x + 2 \right)} - 5 + \frac{5}{2 x + 1} \]
   algebraSimplify the fraction (20*x)/(4*x + 2) by dividing numerator and denominator by 2.✓ Proved
10. \[ = 5 \ln{\left(4 x + 2 \right)} - 5 + \frac{10 x + 5}{2 x + 1} \]
    algebra algebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
11. \[ = 5 \ln{\left(4 x + 2 \right)} \]
    algebra simplifySimplify the fraction (2*x + 1)/(2*x + 1) to 1. Final simplification.✓ Proved