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

Problem 2.1208 · hard

1. \[ \frac{d}{d x} \left(- x^{2} + 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} \]
   sum constant-multipleApply the sum rule. Factor out the constant 1/2.✓ Proved
3. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} \]
   constantMove the constant outside the derivative.✓ Proved
4. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{\frac{d}{d x} \left(2 x + 1\right)}{2 \left(2 x + 1\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
5. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{1}{2 x + 1} \]
   derivative algebraDifferentiate the inner function 2x + 1. Simplify the constant term.✓ Proved
6. \[ = 2 x \left(x + 1\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} 2 x \left(x + 1\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{1}{2 x + 1} \]
   productApply the product rule to the second term.✓ Proved
7. \[ = 2 x \left(x + 1\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + 2 x\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{1}{2 x + 1} \]
   algebraExpand the polynomial inside the derivative.✓ Proved
8. \[ = \frac{2 x \left(x + 1\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + 2 x\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{1}{2 x + 1} \]
   chainApply the chain rule to the logarithm term again.✓ Proved
9. \[ = \frac{4 x \left(x + 1\right)}{2 x + 1} + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + 2 x\right) + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{1}{2 x + 1} \]
   derivative algebraDifferentiate the inner function 2x + 1. Simplify the product of constants and variables.✓ Proved
10. \[ = \frac{4 x \left(x + 1\right)}{2 x + 1} - 2 x + \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + 2 x\right) - 1 + \frac{1}{2 x + 1} \]
    derivativeDifferentiate the remaining polynomial terms.✓ Proved
11. \[ = \frac{4 x \left(x + 1\right)}{2 x + 1} - 2 x + \left(4 x + 2\right) \ln{\left(2 x + 1 \right)} - 1 + \frac{1}{2 x + 1} \]
    derivativeDifferentiate the polynomial 2x^2 + 2x.✓ Proved
12. \[ = - 2 x + \left(4 x + 2\right) \ln{\left(2 x + 1 \right)} - 1 + \frac{4 x^{2} + 4 x + 1}{2 x + 1} \]
    algebraCombine the fractions with the same denominator.✓ Proved
13. \[ = \left(4 x + 2\right) \ln{\left(2 x + 1 \right)} \]
    algebra simplify simplifyRecognize the numerator as a perfect square. Simplify the expression by canceling terms. Final simplified result.✓ Proved