The derivative of ₈≤ft( ₅ x) w. r. t. x is

Correct Answer is : 1xlog⁡8log⁡x1/x 8 xxlog8logx1

Correct Answer is : 1xlog⁡8log⁡x1/x 8 xxlog8logx1

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MHTCET 16 Apr 2026 Shift 1 Paper

Question : 121 of 150

Marks: +1, -0

\log _{8}\left(\log _{5} x\right)

x

Solution:

Concept:

Use the chain rule for a logarithm with any base:

\frac{d}{dx}\log_a u = \frac{u'}{u \log a}

Explanation:

Let

y = \log_8(\log_5 x)

Set

u = \log_5 x

. Then

y = \log_8 u

Differentiate the outer logarithm:

\frac{dy}{dx} = \frac{1}{u \log 8} \cdot \frac{du}{dx}

Now differentiate

u = \log_5 x

\frac{du}{dx} = \frac{1}{x \log 5}

Substitute

u = \log_5 x

and

\frac{du}{dx}

into the derivative:

\frac{dy}{dx} = \frac{1}{\log_5 x \log 8} \cdot \frac{1}{x \log 5}

Since

\log_5 x = \frac{\log x}{\log 5}

, the

\log 5

factors cancel:

\frac{dy}{dx} = \frac{1}{x \log 8 \log x}

Answer:

The correct option is D:

\frac{1}{x \log 8 \log x}

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