if y=x^{( x)} ; then \;dy/dx is

Correct Answer is : yx[log⁡(log⁡x)+1]\;y/x[( x)+1]xy[log(logx)+1]

Correct Answer is : yx[log⁡(log⁡x)+1]\;y/x[( x)+1]xy[log(logx)+1]

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Differential Calculus

Question : 34 of 69

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y=x^{\log(\log x)} ;

\;\frac{dy}{dx}

Solution:

Concept:

Logarithmic differentiation is used for functions of the form

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

Explanation:

Given

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

Take natural log:

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

Differentiate implicitly:

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

Simplify:

\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} \bigl( \log(\log x) + 1 \bigr)

Multiply by

y

\frac{dy}{dx} = \frac{y}{x} \bigl( \log(\log x) + 1 \bigr)

Answer:

\frac{dy}{dx} = \frac{y}{x}\bigl(\log(\log x) + 1\bigr)

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