Derivative of \( \displaystyle 5 x \ln{\left(3 x + 2 \right)} - 5 x + \frac{10 \ln{\left(3 x + 2 \right)}}{3} \)

Problem 2.1230 · hard

1. \[ \frac{d}{d x} \left(5 x \ln{\left(3 x + 2 \right)} - 5 x + \frac{10 \ln{\left(3 x + 2 \right)}}{3}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{10 \ln{\left(3 x + 2 \right)}}{3} \]
   sumApply the sum rule for differentiation.✓ Proved
3. \[ = \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{10 \ln{\left(3 x + 2 \right)}}{3} - 5 \]
   constantDifferentiate the term -5*x.✓ Proved
4. \[ = \frac{d}{d x} 5 x \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
   constant-multipleApply the constant multiple rule to the third term.✓ Proved
5. \[ = 5 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 5 \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 5 \ln{\left(3 x + 2 \right)} + \frac{10 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} - 5 \]
   derivative algebraDifferentiate x. Distribute the 5.✓ Proved
7. \[ = \left(5 x + \frac{10}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 5 \ln{\left(3 x + 2 \right)} - 5 \]
   algebraGroup the terms involving the derivative of the logarithm.✓ Proved
8. \[ = 5 \ln{\left(3 x + 2 \right)} - 5 + \frac{3 \left(5 x + \frac{10}{3}\right)}{3 x + 2} \]
   derivative algebraDifferentiate the logarithm using the chain rule. Simplify the derivative of the logarithm.✓ Proved
9. \[ = 5 \ln{\left(3 x + 2 \right)} - 5 + \frac{15 x + 10}{3 x + 2} \]
   algebraMultiply the term by the fraction.✓ Proved
10. \[ = 5 \ln{\left(3 x + 2 \right)} \]
    algebra algebra algebra simplifyCombine the terms over a common denominator. Distribute the -5 in the numerator. Simplify the numerator. Final simplification.✓ Proved