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

Problem 2.286 · medium

1. \[ \frac{d}{d x} x \left(\ln{\left(x \right)} - 1\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(\ln{\left(x \right)} - 1\right) + \left(\ln{\left(x \right)} - 1\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = x \frac{d}{d x} \left(\ln{\left(x \right)} - 1\right) + \ln{\left(x \right)} - 1 \]
   derivative constant-multipleDifferentiate the first part of the product. Simplify the first term.✓ Proved
4. \[ = x \left(- \frac{d}{d x} 1 + \frac{d}{d x} \ln{\left(x \right)}\right) + \ln{\left(x \right)} - 1 \]
   sumDifferentiate the second part of the sum.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} - 1 \]
   derivative algebraDifferentiate the constant 1. Simplify the expression.✓ Proved
6. \[ = \ln{\left(x \right)} \]
   derivative algebra simplifyDifferentiate the natural logarithm. Simplify the product of x and 1/x. Combine the remaining terms.✓ Proved