Step 1: Understanding the Question:
This problem requires calculating the first and second derivatives of the function \( x^x \) and evaluating their combination at \( x = 2 \).
The function \( x^x \) is differentiated using logarithmic differentiation.
Step 2: Key Formula or Approach:
1. Derivative of \( x^x \): \( y' = x^x (1 + \ln x) \).
2. Second derivative using product rule.
Step 3: Detailed Explanation:
Finding \( y' \):
Let \( y = x^x \). Then \( \ln y = x \ln x \).
Differentiating: \( \frac{1}{y} y' = 1 \cdot \ln x + x \cdot \frac{1}{x} = 1 + \ln x \).
So, \( y' = x^x(1 + \ln x) \).
At \( x = 2 \): \( y'(2) = 2^2(1 + \ln 2) = 4(1 + \ln 2) = 4 + 4 \ln 2 \).
Finding \( y'' \):
Differentiate \( y' = x^x(1 + \ln x) \) using product rule:
\[ y'' = \frac{d}{dx}(x^x) \cdot (1 + \ln x) + x^x \cdot \frac{d}{dx}(1 + \ln x) \]
\[ y'' = x^x(1 + \ln x)^2 + x^x \left(\frac{1}{x}\right) \]
At \( x = 2 \):
\[ y''(2) = 2^2(1 + \ln 2)^2 + 2^2 \left(\frac{1}{2}\right) \]
\[ y''(2) = 4(1 + 2\ln 2 + (\ln 2)^2) + 2 = 4 + 8\ln 2 + 4(\ln 2)^2 + 2 \]
\[ y''(2) = 6 + 8\ln 2 + 4(\ln 2)^2 \]
Evaluating the expression \( y''(2) - 2y'(2) \):
\[ (6 + 8\ln 2 + 4(\ln 2)^2) - 2(4 + 4\ln 2) \]
\[ 6 + 8\ln 2 + 4(\ln 2)^2 - 8 - 8\ln 2 \]
\[ 4(\ln 2)^2 - 2 \]
Step 4: Final Answer:
The result is \( 4(\log_e 2)^2 - 2 \).