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

Problem 2.238 · hard

1. \[ \frac{d}{d x} \frac{5 x^{2} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} x^{2} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} \]
   constant-multiplePull out the constant factor 5/4.✓ Proved
3. \[ = \frac{5 x^{2} \frac{d}{d x} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} + \frac{5 \left(2 \ln{\left(5 x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]
   productApply the product rule.✓ Proved
4. \[ = \frac{5 x^{2} \frac{d}{d x} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} + \frac{5 x \left(2 \ln{\left(5 x \right)} - 1\right)}{2} \]
   derivativeDifferentiate x**2.✓ Proved
5. \[ = \frac{5 x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(5 x \right)}\right)}{4} + \frac{5 x \left(2 \ln{\left(5 x \right)} - 1\right)}{2} \]
   sumApply the sum rule to the second term.✓ Proved
6. \[ = \frac{5 x^{2} \frac{d}{d x} \ln{\left(5 x \right)}}{2} + \frac{5 x \left(2 \ln{\left(5 x \right)} - 1\right)}{2} \]
   derivativeDifferentiate the terms inside the parenthesis.✓ Proved
7. \[ = \frac{5 x \left(2 \ln{\left(5 x \right)} - 1\right)}{2} + \frac{5 x}{2} \]
   chain algebra algebraApply the chain rule to log(5*x). Simplify the derivative of the logarithm. Simplify the term x**2 * (2/x).✓ Proved
8. \[ = 5 x \ln{\left(5 x \right)} \]
   algebra simplify simplifyDistribute 2*x into the parenthesis. Combine like terms. Multiply the remaining terms to get the final answer.✓ Proved