If $f(x)=\ln(\ln x)$, then $ \frac{{\rm d}{f}}{{\rm d}x} =$

Question

BlackTom AI · Accepted Answer

The function is f(x) = ln(ln x). To differentiate, apply the chain rule twice.

First, think of f as ln(u) with u = ln x. The derivative of ln(u) with respect to u is 1/u, so df/dx = (1/ln x) * d/dx(ln x).

Next, d/dx(ln x) = 1/x. Multiplying these factors gives df/dx = (1/ln x) * (1/x) = 1/(x ln x).

Important notes:
- This derivative is valid for x > 1, since ln x must be defined and positive inside the outer ln for real-valued result.
- A common mistake is to omit the inner derivative 1/x from the chain rule or to misplace the reciprocal, which would yield an incorrect expression.

Regarding the provided answer set: the options were not supplied here, so a direct comparison cannot be made. Nevertheless, the derived result 1/(x ln x) is the standard derivative obtained via straightforward application of the chain rule to ln(ln x).

If \(f(x)=\ln(\ln x)\), then \( \frac{{\rm d}{f}}{{\rm d}x} =\)单项选择题

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