Calculus
# Higher-order Derivatives

What is the third derivative of \( \large x^{x^x} \) at \(x=1?\)

Compute \( m+n\), where \( m\) and \( n\) are relatively prime, where

\[\begin{equation} \DeclareMathOperator*{sinc}{sinc\,} \left.\frac{d^{100}}{dx^{100}}(\sinc x)\right|_{x=0}=\frac{m}{n}. \end{equation}\]

For some infinitely differentiable function \(f(x) \), it satisfies the above equation. Given the initial conditions

\[ f(1) = 6, f'(1) = 5, f''(1) = -3 \]

What is the value of \( f( \ln(42) ) \)?

**Details and Assumptions**

\( f'(k), f''(k), f'''(k) \) denote the first, second, and third derivative of \(f(x) \) at \(x=k\) respectively.

Let \(f^{(n)}(x)\) be defined as the \(n\)-th derivative of \(\frac{\ln(x)}{x}\).

If \(f^{(n)}(x)\) can be written in the form shown above, then the solution to \(f^{(n)}(x)=0\) can be written in the form \(x=e^{\frac{p_n}{q_n}}\) where \(p_n\) and \(q_n\) are coprime positive integers.

What is \(p_{10}+q_{10}\)?

\[ \LARGE x^{x^{x^{x^{x^{x^x}}}}}\]

Find the third derivative of the function above at \(x=1\).

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