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

Problem 2.1371 · hard

1. \[ \frac{d}{d x} \left(5 x \ln{\left(2 x + 1 \right)} - 5 x + \frac{5 \ln{\left(2 x + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} \]
   sumApply the sum rule for derivatives.✓ Proved
3. \[ = \frac{d}{d x} 5 x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constantThe derivative of 5*x is 5.✓ Proved
4. \[ = 5 \frac{d}{d x} x \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{5 \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constant-multipleFactor out the common constant 5.✓ Proved
5. \[ = 5 \frac{d}{d x} x \ln{\left(2 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   constant-multipleFactor out 5 from the third term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 5 \ln{\left(2 x + 1 \right)} \frac{d}{d x} x + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   productApply the product rule to the first term.✓ Proved
7. \[ = 5 x \frac{d}{d x} \ln{\left(2 x + 1 \right)} + 5 \ln{\left(2 x + 1 \right)} + \frac{5 \frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 5 \]
   derivative algebraThe derivative of x is 1. Simplify the coefficient of the first term.✓ Proved
8. \[ = \frac{5 x \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} + 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{5 \frac{d}{d x} \left(2 x + 1\right)}{2 \left(2 x + 1\right)} \]
   chainApply the chain rule to the log terms.✓ Proved
9. \[ = \frac{10 x}{2 x + 1} + 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{5}{2 x + 1} \]
   derivative algebra algebraThe derivative of 2*x + 1 is 2. Simplify the expressions inside the parentheses. Distribute the 5.✓ Proved
10. \[ = 5 \ln{\left(2 x + 1 \right)} - 5 + \frac{10 x + 5}{2 x + 1} \]
    algebra algebraCombine the fractions. Factor out 5 from the numerator.✓ Proved
11. \[ = 5 \ln{\left(2 x + 1 \right)} \]
    algebra simplifySimplify the fraction. Final simplification.✓ Proved