Derivative of \( \displaystyle x \left(\ln{\left(5 x \right)} - 1\right) \)

Problem 2.162 · hard

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