Derivative of \( \displaystyle x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \)

Problem 2.1116 · hard

1. \[ \frac{d}{d x} x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(6 \ln{\left(4 x \right)} - 3\right) + \left(6 \ln{\left(4 x \right)} - 3\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(6 \ln{\left(4 x \right)} - 3\right) + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(- \frac{d}{d x} 3 + \frac{d}{d x} 6 \ln{\left(4 x \right)}\right) + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]
   sumDistribute the derivative over the subtraction.✓ Proved
5. \[ = 6 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]
   derivative constantDifferentiate the terms inside the parentheses. The derivative of the constant 3 is 0.✓ Proved
6. \[ = 2 x \left(6 \ln{\left(4 x \right)} - 3\right) + \frac{3 x \frac{d}{d x} 4 x}{2} \]
   chainApply the chain rule to log(4*x).✓ Proved
7. \[ = 2 x \left(6 \ln{\left(4 x \right)} - 3\right) + 6 x \]
   derivative algebra algebraDifferentiate 4*x. Simplify the expression inside the second term. Simplify the product of x**2 and 6/x.✓ Proved
8. \[ = 12 x \ln{\left(4 x \right)} \]
   algebra simplifyDistribute 2*x into the parentheses. Combine like terms.✓ Proved