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

Problem 2.985 · medium

1. \[ \frac{d}{d x} 2 x \left(\ln{\left(4 x \right)} - 1\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = 2 \frac{d}{d x} x \left(\ln{\left(4 x \right)} - 1\right) \]
   constant-multiplePull out the constant factor 2.✓ Proved
3. \[ = 2 x \frac{d}{d x} \left(\ln{\left(4 x \right)} - 1\right) + 2 \left(\ln{\left(4 x \right)} - 1\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 2 x \frac{d}{d x} \left(\ln{\left(4 x \right)} - 1\right) + 2 \ln{\left(4 x \right)} - 2 \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = 2 x \left(- \frac{d}{d x} 1 + \frac{d}{d x} \ln{\left(4 x \right)}\right) + 2 \ln{\left(4 x \right)} - 2 \]
   sumDistribute the derivative over the subtraction.✓ Proved
6. \[ = 2 x \frac{d}{d x} \ln{\left(4 x \right)} + 2 \ln{\left(4 x \right)} - 2 \]
   constantThe derivative of the constant 1 is 0.✓ Proved
7. \[ = 2 \ln{\left(4 x \right)} + \frac{\frac{d}{d x} 4 x}{2} - 2 \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = 2 \ln{\left(4 x \right)} \]
   derivative algebra algebra simplifyDifferentiate the inner function 4*x. Simplify the fraction. Simplify the product of x and 1/x. Combine the constant terms inside the parentheses.✓ Proved