If Y=X^{x} then \;{d² y}{d x²}=

Correct Answer is : dydx(1+log⁡x)+y ddx(1+log⁡x)\;d y/d x(1+ x)+y \;d/d x(1+ x)dxdy(1+logx)+ydxd(1+logx)

\;d y/d x(1+ x)+y \;d/d x(1+ x)

\;d y/d x(1+ x)-y \;d/d x(1+ x)

Correct Answer is : dydx(1+log⁡x)+y ddx(1+log⁡x)\;d y/d x(1+ x)+y \;d/d x(1+ x)dxdy(1+logx)+ydxd(1+logx)

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Differential Calculus

Question : 28 of 69

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Y=X^{x}

\;\frac{d^{2} y}{d x^{2}}=

Solution:

Concept:

Use logarithmic differentiation to find the first derivative, then differentiate again.

Explanation:

Given

y = x^x

Take

\ln y = x \ln x

Differentiate:

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

, so

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

Now differentiate

\frac{dy}{dx}

\frac{d^2 y}{dx^2} = \frac{d}{dx} \left[ y(1+\ln x) \right] = \frac{dy}{dx}(1+\ln x) + y \cdot \frac{d}{dx}(1+\ln x)

Answer:

Option A:

\frac{d y}{d x}(1+\log x)+y \frac{d}{d x}(1+\log x)

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