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JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Differentiation

JEE Main · Mathematics · Differentiation

If h(x)=x^ x^x , then at x=1, h^ (x) h(x) is equal to

If

h(x)=x^{x^x}

, then at

x=1, \frac{h^{\prime}(x)}{h(x)}

is equal to

Options

A $h(x)$
B $\frac{1}{h(x)}$
C $1+\log h(x)$
D $-\log h(x)$

Correct answer

C. $1+\log h(x)$

Step-by-step solution

We have,

\begin{aligned} h(x) & =x^{x^{x^*}} \\ \Rightarrow \quad \log h(x) & =x^x \log x\end{aligned}

On differentiating w. r. t.

x

, we get as

\begin{aligned} \frac{h^{\prime}(x)}{h(x)} & =x^x \cdot \frac{1}{x}+x^x(1+\log x) \log x \\ & =x^{x-1}+(1+\log x) \log h(x)[\text { from Eq. (i) }] \\ & =1+(1+\log 1) \log h(x) \\ & =1+\log h(x)\end{aligned}

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