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

Problem 2.1779 · hard

1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} + \frac{x}{2} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)}\right) \]
   sumStart with the derivative of the entire expression.✓ Proved
2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{2} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   sumApply the sum rule to the derivative of the polynomial.✓ Proved
5. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   constantThe derivative of the constant 1/2 is 0.✓ Proved
6. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} x - \ln{\left(x + 1 \right)} \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   algebra algebraDistribute the log(x + 1) term. Remove the zero term.✓ Proved
7. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
   algebraFactor out log(x + 1) from the parenthesis.✓ Proved
8. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
   logarithmicDifferentiate the log(x + 1) term.✓ Proved
9. \[ = \frac{x}{2} - \left(x + 1\right) \ln{\left(x + 1 \right)} + \frac{1}{2} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
   power algebra algebraDifferentiate the power terms. Simplify the coefficients. Factor out 1/2 from the numerator.✓ Proved
10. \[ = - \left(x + 1\right) \ln{\left(x + 1 \right)} \]
    algebra algebra algebraRecognize the perfect square trinomial. Cancel the common factor (x + 1). Distribute the division by 2.✓ Proved
11. \[ = \left(- x - 1\right) \ln{\left(x + 1 \right)} \]
    simplifyCombine all remaining terms.✓ Proved