Derivative of \( \displaystyle \ln{\left(\frac{e^{- 3 x}}{9 x^{2}} \right)} \)

Problem 2.1031 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{e^{- 3 x}}{9 x^{2}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- 3 x - \ln{\left(9 x^{2} \right)}\right) \]
   algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(-3*x)) to -3*x.✓ Proved
3. \[ = \frac{d}{d x} \left(- 3 x - \ln{\left(x^{2} \right)} - \ln{\left(9 \right)}\right) \]
   algebraUse the property log(ab) = log(a) + log(b).✓ Proved
4. \[ = \frac{d}{d x} \left(- 3 x - 2 \ln{\left(x \right)} - \ln{\left(9 \right)}\right) \]
   powerUse the property log(x**n) = n*log(x).✓ Proved
5. \[ = \frac{d}{d x} \left(- 3 x\right) - \frac{d}{d x} \ln{\left(9 \right)} - \frac{d}{d x} 2 \ln{\left(x \right)} \]
   sumApply the derivative to each term separately.✓ Proved
6. \[ = - \frac{d}{d x} 2 \ln{\left(x \right)} - 3 \]
   constantThe derivative of the constant log(9) is 0.✓ Proved
7. \[ = - 2 \frac{d}{d x} \ln{\left(x \right)} - 3 \]
   constant-multiplePull the constant 2 out of the derivative.✓ Proved
8. \[ = -3 - \frac{2}{x} \]
   logarithmic simplifyThe derivative of log(x) is 1/x. Simplify the expression.✓ Proved