If y = x x , then d y d x =

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Question: Answered & Verified by Expert

If

y = x^{\sqrt{x}},

then

\frac{d y}{d x} =

MathematicsDifferentiationTS EAMCETTS EAMCET 2021 (05 Aug Shift 2)

Options:

A $\frac{\ln x}{2 \sqrt{2}}$
B $\frac{x^{\sqrt{x}}}{\sqrt{x}}$
C $\frac{y \ln x}{2 \sqrt{x}}$
D $\frac{y (\ln x + 2)}{2 \sqrt{x}}$

Solution:

1914 Upvotes Verified Answer

The correct answer is:

\frac{y (\ln x + 2)}{2 \sqrt{x}}

Given, $y = x^{\sqrt{x}}$

Taking $\log$ on both sides, we get

$\ln y = \sqrt{x} \ln x$

$\Rightarrow \frac{1}{y} \frac{d y}{d x} = \sqrt{x} \times \frac{1}{x} + \ln x \times \frac{1}{2} \times (x^{\frac{- 1}{2}})$

$\Rightarrow \frac{1}{y} \frac{d y}{d x} = \frac{1}{2 \sqrt{x}} (2 + \ln x)$

$∴ \frac{d y}{d x} = \frac{y (\ln x + 2)}{2 \sqrt{x}}$

Hence, option $(4)$ is correct.

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