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

Problem 2.993 · hard

1. \[ \frac{d}{d x} x^{2} \left(4 \ln{\left(4 x \right)} - 2\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(4 \ln{\left(4 x \right)} - 2\right) + \left(4 \ln{\left(4 x \right)} - 2\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(4 \ln{\left(4 x \right)} - 2\right) + 2 x \left(4 \ln{\left(4 x \right)} - 2\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(- \frac{d}{d x} 2 + \frac{d}{d x} 4 \ln{\left(4 x \right)}\right) + 2 x \left(4 \ln{\left(4 x \right)} - 2\right) \]
   sumDistribute the derivative over the subtraction.✓ Proved
5. \[ = x^{2} \frac{d}{d x} 4 \ln{\left(4 x \right)} + 2 x \left(4 \ln{\left(4 x \right)} - 2\right) \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = 4 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(4 \ln{\left(4 x \right)} - 2\right) \]
   constant-multiplePull the constant 4 out of the derivative.✓ Proved
7. \[ = 2 x \left(4 \ln{\left(4 x \right)} - 2\right) + x \frac{d}{d x} 4 x \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = 2 x \left(4 \ln{\left(4 x \right)} - 2\right) + 4 x \]
   derivative algebra algebra algebraDifferentiate 4*x. Simplify the fraction 4/(4*x). Simplify the expression. Simplify x**2/x.✓ Proved
9. \[ = 8 x \ln{\left(4 x \right)} \]
   algebra simplifyDistribute 2*x. Combine like terms.✓ Proved