Derivative of \( \displaystyle \frac{5 x^{2} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} \)

Problem 2.961 · hard

1. \[ \frac{d}{d x} \frac{5 x^{2} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} x^{2} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} \]
   constant-multiplePull out the constant factor 5/4.✓ Proved
3. \[ = \frac{5 x^{2} \frac{d}{d x} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} + \frac{5 \left(1 - 2 \ln{\left(5 x \right)}\right) \frac{d}{d x} x^{2}}{4} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{5 x^{2} \frac{d}{d x} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} + \frac{5 x \left(1 - 2 \ln{\left(5 x \right)}\right)}{2} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = \frac{5 x^{2} \left(\frac{d}{d x} 1 - 2 \frac{d}{d x} \ln{\left(5 x \right)}\right)}{4} + \frac{5 x \left(1 - 2 \ln{\left(5 x \right)}\right)}{2} \]
   sumDifferentiate the second part of the product.✓ Proved
6. \[ = - \frac{5 x^{2} \frac{d}{d x} \ln{\left(5 x \right)}}{2} + \frac{5 x \left(1 - 2 \ln{\left(5 x \right)}\right)}{2} \]
   derivativeDifferentiate the terms inside the parenthesis.✓ Proved
7. \[ = \frac{5 x \left(1 - 2 \ln{\left(5 x \right)}\right)}{2} - \frac{x \frac{d}{d x} 5 x}{2} \]
   derivativeApply the chain rule to the logarithm.✓ Proved
8. \[ = \frac{5 x \left(1 - 2 \ln{\left(5 x \right)}\right)}{2} - \frac{5 x}{2} \]
   derivative algebra algebraDifferentiate the inner function 5*x. Simplify the derivative of the inner function. Simplify the term x**2 * (2/x).✓ Proved
9. \[ = - 5 x \ln{\left(5 x \right)} \]
   algebra simplify simplifyDistribute 2*x into the parenthesis. Combine like terms. Final simplification.✓ Proved