Derivative of \( \displaystyle x^{2} \left(10 \ln{\left(4 x \right)} - 5\right) \)

Problem 2.290 · hard

1. \[ \frac{d}{d x} x^{2} \left(10 \ln{\left(4 x \right)} - 5\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x^{2} \frac{d}{d x} \left(10 \ln{\left(4 x \right)} - 5\right) + \left(10 \ln{\left(4 x \right)} - 5\right) \frac{d}{d x} x^{2} \]
   productApply the product rule.✓ Proved
3. \[ = x^{2} \frac{d}{d x} \left(10 \ln{\left(4 x \right)} - 5\right) + 2 x \left(10 \ln{\left(4 x \right)} - 5\right) \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = x^{2} \left(- \frac{d}{d x} 5 + \frac{d}{d x} 10 \ln{\left(4 x \right)}\right) + 2 x \left(10 \ln{\left(4 x \right)} - 5\right) \]
   sumDistribute the derivative over the subtraction.✓ Proved
5. \[ = 10 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(10 \ln{\left(4 x \right)} - 5\right) \]
   constant constant-multipleDifferentiate the constant 5. Factor out the constant 10.✓ Proved
6. \[ = 2 x \left(10 \ln{\left(4 x \right)} - 5\right) + \frac{5 x \frac{d}{d x} 4 x}{2} \]
   chainApply the chain rule to the logarithm.✓ Proved
7. \[ = 2 x \left(10 \ln{\left(4 x \right)} - 5\right) + 10 x \]
   derivative algebra algebraDifferentiate the inner function 4*x. Simplify the expression inside the second term. Simplify the product of x**2 and 10/x.✓ Proved
8. \[ = 20 x \ln{\left(4 x \right)} \]
   algebra simplifyDistribute 2*x into the parentheses. Combine like terms.✓ Proved