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UPSC NDA I 2022 Mathematics Solved Paper

Question : 70 of 120

Marks: +1, -0

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

\frac{dy}{dx}

x = 1

Solution:

Concept:

Use logarithmic differentiation to find the derivative of a function where the variable appears in both base and exponent.

Explanation:

Given

y = (x^x)^x

Take natural log on both sides:

\ln y = \ln\left((x^x)^x\right) = x \ln(x^x) = x \cdot x \ln x = x^2 \ln x

Differentiate with respect to

x

\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x^2 \ln x)

Apply product rule:

\frac{1}{y} \frac{dy}{dx} = x^2 \cdot \frac{1}{x} + \ln x \cdot 2x = x + 2x \ln x

Thus

\frac{dy}{dx} = y (x + 2x \ln x) = (x^x)^x (x + 2x \ln x)

Now substitute

x = 1

y(1) = (1^1)^1 = 1

\ln 1 = 0

, so

\frac{dy}{dx}\big|_{x=1} = 1 \cdot (1 + 2\cdot 1 \cdot 0) = 1

Answer:

1

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