Derivative of \( \displaystyle x^{2} \left(1 - 2 \ln{\left(4 x \right)}\right) \)

Problem 2.669 · hard

1. \[ \frac{d}{d x} x^{2} \left(1 - 2 \ln{\left(4 x \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(1 - 2 \ln{\left(4 x \right)}\right) + \left(1 - 2 \ln{\left(4 x \right)}\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(1 - 2 \ln{\left(4 x \right)}\right) + 2 x \left(1 - 2 \ln{\left(4 x \right)}\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(\frac{d}{d x} 1 - 2 \frac{d}{d x} \ln{\left(4 x \right)}\right) + 2 x \left(1 - 2 \ln{\left(4 x \right)}\right) \]
   derivativeDifferentiate the second part of the product.✓ Proved
5. \[ = - 2 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(1 - 2 \ln{\left(4 x \right)}\right) \]
   derivativeDifferentiate the constant 1.✓ Proved
6. \[ = 2 x \left(1 - 2 \ln{\left(4 x \right)}\right) - \frac{x \frac{d}{d x} 4 x}{2} \]
   logarithmicApply the chain rule for the logarithm.✓ Proved
7. \[ = 2 x \left(1 - 2 \ln{\left(4 x \right)}\right) - 2 x \]
   derivative algebra algebraDifferentiate the inner function 4*x. Simplify the expression inside the second term. Simplify the product of x**2 and -2/x.✓ Proved
8. \[ = - 4 x \ln{\left(4 x \right)} \]
   algebra simplifyDistribute 2*x into the parentheses. Combine like terms.✓ Proved