Questions: Find the second derivative of the function. f(x) = (ln(x))^2 f''(x) = □

Solution Steps

To find the second derivative of the function \( f(x) = (\ln(x))^2 \), we first need to find the first derivative using the chain rule and the derivative of the natural logarithm. Once we have the first derivative, we differentiate it again to find the second derivative.

To find the first derivative of \( f(x) = (\ln(x))^2 \), we use the chain rule. The derivative of \((\ln(x))^2\) is: \[ f'(x) = 2 \cdot \ln(x) \cdot \frac{1}{x} = \frac{2 \ln(x)}{x} \]

Next, we differentiate \( f'(x) = \frac{2 \ln(x)}{x} \) to find the second derivative. Using the quotient rule, we have: \[ f''(x) = \frac{d}{dx} \left( \frac{2 \ln(x)}{x} \right) = \frac{x \cdot \frac{2}{x} - 2 \ln(x) \cdot 1}{x^2} = \frac{2 - 2 \ln(x)}{x^2} \]

Final Answer