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MHTCET 15 Apr 2026 Shift 1 Paper

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Question : 120 of 150

Marks: +1, -0

If

y=\log x^{x}

, then the value of

\left(\frac{d y}{d x}\right)^{2}

x=e^{2}

is

4
e
$e^{2}$
9

Solution:

Concept:

Use the logarithm power rule to simplify, then differentiate using the product rule.

Explanation:

Given

y = \log x^x

.

Apply

\log a^m = m \log a

, so

y = x \log x

.

Differentiate with respect to

x

using the product rule:

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

.

Thus,

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

.

At

x = e^2

, we have

\log e^2 = 2

.

So

\frac{dy}{dx} = 1 + 2 = 3

.

Hence,

\left(\frac{dy}{dx}\right)^2 = 3^2 = 9

.

Answer:

Option D:

9

.

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