If x^{y}=e²(x-y)}, then \;dy/dx is equal to

Correct Answer is : 2(1+log⁡x)(2+log⁡x)2\;2(1+ x)/(2+ x)²(2+logx)22(1+logx)

Correct Answer is : 2(1+log⁡x)(2+log⁡x)2\;2(1+ x)/(2+ x)²(2+logx)22(1+logx)

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KEAM 2010 Math Paper

Question : 28 of 120

Marks: +1, -0

x^{y}=e^{2(x-y)},

\;\frac{dy}{dx}

is equal to

Solution:

We have

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

Taking log on both sides, we have

\log (x^{y})=\log (e^{2(x-y)})

\Rightarrow \;y \log x=2(x-y)

\Rightarrow \;y=\;\frac{2x}{\log x+z}

...(i)

Differentiating w.r.t.

x

, we have

y \cdot \;\frac{1}{x}+\;\frac{dy}{dx} \log x=2\left(1-\;\frac{dy}{dx}\right)

\Rightarrow \;\frac{dy}{dx}(\log x+2)=2-\;\frac{y}{x}

\Rightarrow \;\frac{dy}{dx}=\;\frac{2x-y}{x(2+\log x)}

\Rightarrow \;\frac{dy}{dx}=\;\frac{2x-\frac{2x}{2+\log x}}{x(2+\log x)}

[Using Eq (i)]

=\;\frac{2x(1+\log x)}{x(2+\log x)^2}

=\;\frac{2(1+\log x)}{(2+\log x)^2}

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