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

Problem 2.387 · hard

1. \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(2 x \right)} - 1\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + \left(2 \ln{\left(2 x \right)} - 1\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
   powerDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(2 x \right)}\right) + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
   sumApply the sum rule to the second part.✓ Proved
5. \[ = 2 x^{2} \frac{d}{d x} \ln{\left(2 x \right)} + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
   constant constant-multipleDifferentiate the constant term. Factor out the constant 2.✓ Proved
6. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + x \frac{d}{d x} 2 x \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \]
   derivative algebra algebraDifferentiate the inner function 2*x. Simplify the expression inside the parentheses. Simplify the second term.✓ Proved
8. \[ = 4 x \ln{\left(2 x \right)} \]
   algebra algebra algebraFactor out 2*x. Simplify the expression inside the parentheses. Final simplification.✓ Proved