If y= ₃≤ft( ₃ x) then d y/d x at x=3 is ......

Correct Answer is : 13(log⁡3)−2 1/3( 3)^{-2}31(log3)−2

Correct Answer is : 13(log⁡3)−2 1/3( 3)^{-2}31(log3)−2

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MHTCET 25 Apr 2025 Shift 1 Paper

Question : 149 of 150

Marks: +1, -0

y=\log _{3}\left(\log _{3} x\right)

\frac{d y}{d x}

x=3

is ......

Solution:

Concept:

Use the change of base formula for logarithms, then differentiate using the chain rule.

Explanation:

Given

y = \log_{3}(\log_{3} x)

Write

\log_{3} x = \frac{\log x}{\log 3}

, so:

y = \frac{\log\left(\frac{\log x}{\log 3}\right)}{\log 3}

y = \frac{\log(\log x) - \log(\log 3)}{\log 3}

Differentiate with respect to

x

\frac{dy}{dx} = \frac{1}{\log 3} \cdot \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x \log 3}

x = 3

, we get:

\left(\frac{dy}{dx}\right)_{x=3} = \frac{1}{3 \log 3 \log 3} = \frac{1}{3(\log 3)^{2}}

Therefore:

\frac{dy}{dx} = \frac{1}{3}(\log 3)^{-2}

Answer:

Option D:

\frac{1}{3}(\log 3)^{-2}

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