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).

SCNC1111_2B_2024 Quiz 6

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

查看解析

登录即可查看完整答案

类似问题

If 𝑦 = ln ⁡ ( 5 𝑥 − 1 ) 4 then 𝑑 𝑦 𝑑 𝑥 =

Find the velocity at t = 0.69 using the following equation defining position. p(t) = 2t3 - 4t2 +11t -2 Round to 2 decimal places.

Differentiate [ ln ⁡ ( 3 𝑥 ) + 5 ] 2

Differentiate cos 3 ⁡ 4 𝑥

Differentiate 𝑒 3 𝑥 2 + 5 𝑥

Differentiate ln ⁡ ( 𝑥 2 − 5 2 𝑥 + 1 )

Differentiate ln ⁡ ( 7 𝑥 + 5 )

Differentiate cos ⁡ ( 3 𝑥 )

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单