2. $$ \left(\frac{\ln (x)}{x^{2}}\right)^{\prime \prime \prime} $$

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We need to calculate the third derivative of the function $$ \frac{\ln (x)}{x^{2}} $$. This involves first finding the first derivative, then the second derivative, and finally the third derivative.
Let's find the first derivative of $$ \frac{\ln (x)}{x^{2}} $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$\frac{d}{dx}\left(\frac{\ln{\left(xight)}}{x^{2}}ight)$$
- step1: Use differentiation rules:
  $$\frac{\frac{d}{dx}\left(\ln{\left(xight)}ight)	imes x^{2}-\ln{\left(xight)}	imes \frac{d}{dx}\left(x^{2}ight)}{\left(x^{2}ight)^{2}}$$
- step2: Find the derivative:
  $$\frac{\frac{1}{x}	imes x^{2}-\ln{\left(xight)}	imes 2x}{\left(x^{2}ight)^{2}}$$
- step3: Multiply the terms:
  $$\frac{x-2x\ln{\left(xight)}}{x^{4}}$$
- step4: Factor:
  $$\frac{x\left(1-2\ln{\left(xight)}ight)}{x^{4}}$$
- step5: Reduce the fraction:
  $$\frac{1-2\ln{\left(xight)}}{x^{3}}$$
Now, we will find the second derivative of the first derivative $$ \frac{1-2\ln{(x)}}{x^{3}} $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$\frac{d}{dx}\left(\frac{1-2\ln{\left(xight)}}{x^{3}}ight)$$
- step1: Use differentiation rules:
  $$\frac{\frac{d}{dx}\left(1-2\ln{\left(xight)}ight)	imes x^{3}-\left(1-2\ln{\left(xight)}ight)	imes \frac{d}{dx}\left(x^{3}ight)}{\left(x^{3}ight)^{2}}$$
- step2: Calculate:
  $$\frac{-\frac{2}{x}	imes x^{3}-\left(1-2\ln{\left(xight)}ight)	imes 3x^{2}}{\left(x^{3}ight)^{2}}$$
- step3: Reduce the fraction:
  $$\frac{-2x^{2}-\left(3x^{2}-6\ln{\left(xight)}	imes x^{2}ight)}{x^{6}}$$
- step4: Subtract the terms:
  $$\frac{-5x^{2}+6\ln{\left(xight)}	imes x^{2}}{x^{6}}$$
- step5: Factor:
  $$\frac{x^{2}\left(-5+6\ln{\left(xight)}ight)}{x^{6}}$$
- step6: Reduce the fraction:
  $$\frac{-5+6\ln{\left(xight)}}{x^{4}}$$
Next, we will find the third derivative of the second derivative $$ \frac{-5+6\ln{(x)}}{x^{4}} $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$\frac{d}{dx}\left(\frac{-5+6\ln{\left(xight)}}{x^{4}}ight)$$
- step1: Use differentiation rules:
  $$\frac{\frac{d}{dx}\left(-5+6\ln{\left(xight)}ight)	imes x^{4}-\left(-5+6\ln{\left(xight)}ight)	imes \frac{d}{dx}\left(x^{4}ight)}{\left(x^{4}ight)^{2}}$$
- step2: Calculate:
  $$\frac{\frac{6}{x}	imes x^{4}-\left(-5+6\ln{\left(xight)}ight)	imes 4x^{3}}{\left(x^{4}ight)^{2}}$$
- step3: Reduce the fraction:
  $$\frac{6x^{3}-\left(-20x^{3}+24\ln{\left(xight)}	imes x^{3}ight)}{x^{8}}$$
- step4: Subtract the terms:
  $$\frac{26x^{3}-24\ln{\left(xight)}	imes x^{3}}{x^{8}}$$
- step5: Factor:
  $$\frac{x^{3}\left(26-24\ln{\left(xight)}ight)}{x^{8}}$$
- step6: Reduce the fraction:
  $$\frac{26-24\ln{\left(xight)}}{x^{5}}$$
The third derivative of the function $$ \frac{\ln (x)}{x^{2}} $$ is $$ \frac{26-24\ln{(x)}}{x^{5}} $$.

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