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

Problem 2.947 · hard

1. \[ \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)} + x - \frac{2 \ln{\left(5 x + 2 \right)}}{5}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) + \frac{d}{d x} \left(- \frac{2 \ln{\left(5 x + 2 \right)}}{5}\right) \]
   sumApply the sum rule to separate the terms.✓ Proved
3. \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) - \frac{d}{d x} \frac{2 \ln{\left(5 x + 2 \right)}}{5} \]
   algebraRewrite the subtraction as addition of a negative term.✓ Proved
4. \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) - \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} \]
   constant-multiplePull the constant factor out of the derivative.✓ Proved
5. \[ = \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) - \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} + 1 \]
   derivativeDifferentiate the term x.✓ Proved
6. \[ = \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) + 1 - \frac{2 \frac{d}{d x} \left(5 x + 2\right)}{5 \left(5 x + 2\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = \frac{d}{d x} \left(- x \ln{\left(5 x + 2 \right)}\right) + 1 - \frac{2}{5 x + 2} \]
   derivative algebraDifferentiate the inner linear function. Simplify the product of the constants and the fraction.✓ Proved
8. \[ = - x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} \left(- x\right) + 1 - \frac{2}{5 x + 2} \]
   productApply the product rule to the first term.✓ Proved
9. \[ = - \frac{x \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} - \ln{\left(5 x + 2 \right)} + 1 - \frac{2}{5 x + 2} \]
   derivativeDifferentiate -x.✓ Proved
10. \[ = - \frac{5 x}{5 x + 2} - \ln{\left(5 x + 2 \right)} + 1 - \frac{2}{5 x + 2} \]
    derivative algebraDifferentiate the inner part of the chain rule again. Simplify the term involving x.✓ Proved
11. \[ = \frac{- 5 x - 2}{5 x + 2} - \ln{\left(5 x + 2 \right)} + 1 \]
    algebraCombine the fractions with the same denominator.✓ Proved
12. \[ = - \ln{\left(5 x + 2 \right)} \]
    algebra simplifySimplify the fraction (-5x-2)/(5x+2) to -1. Final simplification.✓ Proved