Derivative of \( \displaystyle x \left(1 - \ln{\left(2 x \right)}\right) \)

Problem 2.720 · medium

1. \[ \frac{d}{d x} x \left(1 - \ln{\left(2 x \right)}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(1 - \ln{\left(2 x \right)}\right) + \left(1 - \ln{\left(2 x \right)}\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} \left(1 - \ln{\left(2 x \right)}\right) - \ln{\left(2 x \right)} + 1 \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x \left(\frac{d}{d x} 1 - \frac{d}{d x} \ln{\left(2 x \right)}\right) - \ln{\left(2 x \right)} + 1 \]
   sumDifferentiate the second part using the sum rule.✓ Proved
5. \[ = - x \frac{d}{d x} \ln{\left(2 x \right)} - \ln{\left(2 x \right)} + 1 \]
   derivative constantDifferentiate the constant 1. The derivative of 0 is 0.✓ Proved
6. \[ = - \ln{\left(2 x \right)} - \frac{\frac{d}{d x} 2 x}{2} + 1 \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = - \ln{\left(2 x \right)} \]
   derivative algebra algebra simplifyDifferentiate the inner function 2*x. Simplify the fraction. Distribute x into the second term. Combine the constant terms.✓ Proved