Method of DifferentiationmediumFree

Method of Differentiation — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

f(x)=\log_{x}(\log x)

, then

f'(x)

x=e

\dfrac{1}{e}

correct

1

e

DNone of these

Solution

Step 1: Rewrite

f(x)

using the change of base formula (all logs are natural logs):

f(x)=\log_{x}(\log x)=\dfrac{\log(\log x)}{\log x}

Step 2: Differentiate using the quotient rule. Let

u=\log(\log x)

and

v=\log x

u'=\dfrac{1}{\log x}\cdot\dfrac{1}{x}=\dfrac{1}{x\log x},\qquad v'=\dfrac{1}{x}

f'(x)=\dfrac{u'v-uv'}{v^{2}}=\dfrac{\dfrac{1}{x\log x}\cdot\log x\;-\;\log(\log x)\cdot\dfrac{1}{x}}{(\log x)^{2}}=\dfrac{1-\log(\log x)}{x(\log x)^{2}}

Step 3: Evaluate at

x=e

. Since

\log e=1

and

\log(\log e)=\log 1=0

f'(e)=\dfrac{1-0}{e\cdot 1^{2}}=\dfrac{1}{e}

Correct answer: (1)

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