If y=( x)^{ x}, then d y/d x is equal to

Correct Answer is : (sin⁡x)tan⁡x(1+sec⁡2xlog⁡(sin⁡x))( x)^{ x}≤ft(1+ ² x ( x))(sinx)tanx(1+sec2xlog(sinx))

Correct Answer is : (sin⁡x)tan⁡x(1+sec⁡2xlog⁡(sin⁡x))( x)^{ x}≤ft(1+ ² x ( x))(sinx)tanx(1+sec2xlog(sinx))

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MHTCET 11 May 2024 Shift 2 Paper

Question : 148 of 150

Marks: +1, -0

y=(\sin x)^{\tan x}

\frac{d y}{d x}

is equal to

Solution:

Concept:

Use logarithmic differentiation for functions of the form

f(x)^{g(x)}

Explanation:

Given

y=(\sin x)^{\tan x}

Taking natural logarithm on both sides:

\log y = \tan x \cdot \log(\sin x)

Differentiate both sides with respect to

x

\frac{1}{y}\frac{dy}{dx} = \sec^2 x \cdot \log(\sin x) + \tan x \cdot \frac{1}{\sin x} \cdot \cos x

Since

\tan x \cdot \cot x = 1

, we get:

\frac{1}{y}\frac{dy}{dx} = \sec^2 x \log(\sin x) + 1

Multiply by

y = (\sin x)^{\tan x}

\frac{dy}{dx} = (\sin x)^{\tan x}\left(1 + \sec^2 x \log(\sin x)\right)

Answer:

Option A:

(\sin x)^{\tan x}\left(1+\sec^2 x \log(\sin x)\right)

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