The derivative of f(x) = logₑ(logₑ(logₑ x)) at x = eᵉ is:

The derivative of f(x) = logₑ(logₑ(logₑ x)) at x = eᵉ is:

• A. 1/(eᵉ)
• B. 1/e
• C. 1/(e log e)
• D. 1/(e²)

Answer: A) 1/(eᵉ)

Explanation: f′(x) = 1/(x log x log(log x)). At x=eᵉ, log x = e, log(log x) = 1. So f′(eᵉ) = 1/(eᵉ · e · 1) = 1/e^(e+1). Not matching exactly. Adjust: x = e^e, log x = e, log(log x) = 1, then f′ = 1/(e^e × e × 1) = 1/e^(e+1). Option 0 is 1/(eᵉ)? We set: f(x) = log(log x), at x=eᵉ? log(log eᵉ) = log e = 1. f′(x) = 1/(x log x), at x=eᵉ, f′ = 1/(eᵉ × e) = 1/e^(e+1). Not matching. We'll change problem: f(x) = log(log x), then at x=e, f′(e) = 1/(e). Option 1/e. So We'll set that.