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

Problem 2.2015 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} + \frac{x}{2} - \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)}\right) \]
   sumDifferentiate the sum term by term.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)} \]
   sum✓ Proved
3. \[ = - \left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} - \ln{\left(2 x + 1 \right)} \frac{d}{d x} \left(x^{2} + x + \frac{1}{4}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   productApply the product rule to the last term.✓ Proved
4. \[ = - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} - \left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
   derivativeDifferentiate the polynomial part.✓ Proved
5. \[ = - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\left(x^{2} + x + \frac{1}{4}\right) \frac{d}{d x} \left(2 x + 1\right)}{2 x + 1} \]
   chainApply the chain rule to the logarithm.✓ Proved
6. \[ = - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{2 \left(x^{2} + x + \frac{1}{4}\right)}{2 x + 1} \]
   derivativeDifferentiate the inner function 2x + 1.✓ Proved
7. \[ = - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{2 x^{2} + 2 x + \frac{1}{2}}{2 x + 1} \]
   algebraSimplify the fraction.✓ Proved
8. \[ = x - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{1}{2} - \frac{2 x^{2} + 2 x + \frac{1}{2}}{2 x + 1} \]
   derivative algebraDifferentiate the first two terms. Distribute the negative sign.✓ Proved
9. \[ = x - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{1}{2} - \frac{4 x^{2} + 4 x + 1}{4 x + 2} \]
   algebraMultiply the numerator and denominator by 2.✓ Proved
10. \[ = x - \frac{\left(2 x + 1\right)^{2}}{4 x + 2} - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} + \frac{1}{2} \]
    algebraRecognize the perfect square trinomial.✓ Proved
11. \[ = - \left(2 x + 1\right) \ln{\left(2 x + 1 \right)} \]
    simplify algebraCancel the common factor (2x + 1). Split the fraction.✓ Proved
12. \[ = \left(- 2 x - 1\right) \ln{\left(2 x + 1 \right)} \]
    simplifyCombine all remaining terms.✓ Proved