Derivative of \( \displaystyle 5 x \left(\ln{\left(3 x \right)} - 1\right) \)

Problem 2.703 · medium

1. \[ \frac{d}{d x} 5 x \left(\ln{\left(3 x \right)} - 1\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} x \left(\ln{\left(3 x \right)} - 1\right) \]
   constant-multiplePull out the constant factor 5.✓ Proved
3. \[ = 5 x \frac{d}{d x} \left(\ln{\left(3 x \right)} - 1\right) + 5 \left(\ln{\left(3 x \right)} - 1\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 5 x \frac{d}{d x} \left(\ln{\left(3 x \right)} - 1\right) + 5 \ln{\left(3 x \right)} - 5 \]
   derivative derivativeDifferentiate the first part of the product. Differentiate the first part of the second term.✓ Proved
5. \[ = 5 x \left(- \frac{d}{d x} 1 + \frac{d}{d x} \ln{\left(3 x \right)}\right) + 5 \ln{\left(3 x \right)} - 5 \]
   sumSplit the derivative of the difference.✓ Proved
6. \[ = 5 x \frac{d}{d x} \ln{\left(3 x \right)} + 5 \ln{\left(3 x \right)} - 5 \]
   derivativeThe derivative of the constant 1 is 0.✓ Proved
7. \[ = 5 \ln{\left(3 x \right)} + \frac{5 \frac{d}{d x} 3 x}{3} - 5 \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = 5 \ln{\left(3 x \right)} \]
   derivative algebra algebra simplifyDifferentiate the inner function 3*x. Simplify the fraction. Simplify the product of x and 1/x. Combine the terms to get the final answer.✓ Proved