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

Problem 2.2127 · hard

1. \[ \frac{d}{d x} \left(\frac{5 x^{2}}{4} + x - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{4} - \frac{d}{d x} \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) + \frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{4} \]
   productApply the product rule to the third term.✓ Proved
4. \[ = - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} - \left(\frac{d}{d x} \frac{2}{5} + \frac{d}{d x} 2 x + \frac{d}{d x} \frac{5 x^{2}}{2}\right) \ln{\left(5 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{4} \]
   sumApply the sum rule to the first part of the product rule.✓ Proved
5. \[ = - \left(\frac{d}{d x} 2 x + \frac{d}{d x} \frac{5 x^{2}}{2}\right) \ln{\left(5 x + 2 \right)} - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{4} \]
   constant sumThe derivative of the constant 2/5 is 0. Apply the sum rule to the inner terms.✓ Proved
6. \[ = - \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{5 x^{2}}{4} + 1 \]
   derivativeDifferentiate the simple terms.✓ Proved
7. \[ = - \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{5 x^{2}}{4} + 1 - \frac{5 \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right)}{5 x + 2} \]
   chain algebraApply the chain rule to the log term. Simplify the derivative of the log term.✓ Proved
8. \[ = \frac{5 x}{2} - \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} + 1 - \frac{5 \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right)}{5 x + 2} \]
   derivativeDifferentiate the first term.✓ Proved
9. \[ = \frac{5 x}{2} - \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} + 1 - \frac{\frac{25 x^{2}}{2} + 10 x + 2}{5 x + 2} \]
   algebraDistribute the 5 in the last term.✓ Proved
10. \[ = \frac{5 x}{2} - \left(5 x + 2\right) \ln{\left(5 x + 2 \right)} + 1 - \frac{25 x^{2} + 20 x + 4}{10 x + 4} \]
    algebra algebra simplifySimplify the fraction. Expand the denominator. Final simplification of the expression.✓ Proved