Derivative of x^{≤ft(x^{x})} is

Correct Answer is : x(xx)(xx−1+xxlog⁡x(1+log⁡x)) x^{≤ft(x^{x})}≤ft(x^{x-1}+x^{x} x(1+ x))x(xx)(xx−1+xxlogx(1+logx))

x^{≤ft(x^{x})}≤ft(x^{x}+1+ x)

x^{≤ft(x^{x})}≤ft(x^{x}+ x)

x^{≤ft(x^{x})}≤ft(x^{x}+x^{x-1} x(1+ x))

x^{≤ft(x^{x})}≤ft(x^{x-1}+x^{x} x(1+ x))

Correct Answer is : x(xx)(xx−1+xxlog⁡x(1+log⁡x)) x^{≤ft(x^{x})}≤ft(x^{x-1}+x^{x} x(1+ x))x(xx)(xx−1+xxlogx(1+logx))

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MHTCET 22 Apr 2025 Shift 2 Paper

Question : 131 of 150

Marks: +1, -0

Derivative of

x^{\left(x^{x}\right)}

Solution:

Concept:

Use logarithmic differentiation for functions of the form

y = f(x)^{g(x)}

Explanation:

Let

y = x^{\left(x^{x}\right)}

Take natural log on both sides:

\log y = x^{x} \log x

Differentiate both sides with respect to

x

\frac{1}{y}\frac{dy}{dx} = x^{x}\cdot\frac{1}{x} + \log x \cdot \frac{d}{dx}\left(x^{x}\right)

Now differentiate

x^{x}

separately.

Let

t = x^{x}

, so

\log t = x\log x

Differentiating:

\frac{1}{t}\frac{dt}{dx} = 1 + \log x

Thus

\frac{d}{dx}\left(x^{x}\right) = x^{x}(1+\log x)

Substitute back:

\frac{1}{y}\frac{dy}{dx} = x^{x-1} + \log x \cdot x^{x}(1+\log x)

Multiply by

y

\frac{dy}{dx} = x^{\left(x^{x}\right)}\left[x^{x-1} + x^{x}\log x(1+\log x)\right]

Answer:

Option D:

x^{\left(x^{x}\right)}\left(x^{x-1}+x^{x} \log x(1+\log x)\right)

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