If y = (xx)x, then which one of the following is correct ?

Correct Answer is : dydx{{dy}}{{dx}}dxdy − xy(1 + 2 ln x) = 0

{{dy}}{{dx}} − 2xy(1 + ln x) = 0

{{dy}}{{dx}} + 2xy(1 + ln x) = 0

Correct Answer is : dydx{{dy}}{{dx}}dxdy − xy(1 + 2 ln x) = 0

Test Index

UPSC NDA 2 2022 Mathematics Solved Paper

Question : 78 of 120

Marks: +1, -0

y = (xx)x

, then which one of the following is correct ?

Solution:

Concept:

Use logarithmic differentiation to differentiate

y = x^{x^2}

Explanation:

Let

y = (x^x)^x = x^{x^2}

Take natural log:

\ln y = x^2 \ln x

Differentiate both sides with respect to

x

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

Multiply by

y

\frac{dy}{dx} = y(2x \ln x + x) = xy(1 + 2\ln x)

Rearrange:

\frac{dy}{dx} - xy(1 + 2\ln x) = 0

Answer:

Option B:

\frac{dy}{dx} - xy(1 + 2\ln x) = 0

Go to Question:

Prev Question

Next Question