Derivative of \( \displaystyle x \left(\ln{\left(4 x \right)} - 1\right) \)

Problem 2.376 · medium

1. \[ \frac{d}{d x} x \left(\ln{\left(4 x \right)} - 1\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(\ln{\left(4 x \right)} - 1\right) + \left(\ln{\left(4 x \right)} - 1\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} \left(\ln{\left(4 x \right)} - 1\right) + \ln{\left(4 x \right)} - 1 \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = - x \frac{d}{d x} 1 + x \frac{d}{d x} \ln{\left(4 x \right)} + \ln{\left(4 x \right)} - 1 \]
   sumDistribute the derivative across the subtraction.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(4 x \right)} + \ln{\left(4 x \right)} - 1 \]
   derivative algebraDifferentiate the constant term. Simplify the zero term.✓ Proved
6. \[ = \ln{\left(4 x \right)} \]
   chain algebra algebra simplifyApply the chain rule to the logarithm. Simplify the fraction by canceling the 4. Simplify the product of 1/x and x. Combine the remaining terms.✓ Proved