Derivative of \( \displaystyle x \left(1 - \ln{\left(3 x \right)}\right) \)

Problem 2.164 · hard

1. \[ \frac{d}{d x} x \left(1 - \ln{\left(3 x \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(1 - \ln{\left(3 x \right)}\right) + \left(1 - \ln{\left(3 x \right)}\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} \left(1 - \ln{\left(3 x \right)}\right) - \ln{\left(3 x \right)} + 1 \]
   constantThe derivative of x is 1.✓ Proved
4. \[ = x \left(\frac{d}{d x} 1 - \frac{d}{d x} \ln{\left(3 x \right)}\right) - \ln{\left(3 x \right)} + 1 \]
   sumApply the derivative to the sum inside the parentheses.✓ Proved
5. \[ = - x \frac{d}{d x} \ln{\left(3 x \right)} - \ln{\left(3 x \right)} + 1 \]
   constant algebraThe derivative of 1 is 0. Simplify the expression inside the parentheses.✓ Proved
6. \[ = - \ln{\left(3 x \right)} - \frac{\frac{d}{d x} 3 x}{3} + 1 \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = - \ln{\left(3 x \right)} \]
   derivative algebra algebra simplifyThe derivative of 3*x is 3. Simplify the fraction. Multiply x by -1/x. Combine the constant terms.✓ Proved