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y=x^(tan^(-1(x))),Find dy/dx...

y=`x^(tan^(-1_(x)))`,Find dy/dx

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AI Generated Solution

To find the derivative \( \frac{dy}{dx} \) for the function \( y = x^{\tan^{-1}(x)} \), we can use logarithmic differentiation. Here’s the step-by-step solution: ### Step 1: Take the natural logarithm of both sides We start by taking the natural logarithm of both sides to simplify the differentiation process. \[ \ln y = \ln(x^{\tan^{-1}(x)}) \] ...
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Knowledge Check

  • If y=x^(tan ^(-1)x) ,then (dy)/(dx)=

    A
    ` y(( tan ^(-1) x)/( x ) +(logx )/( 1+x^(2)))`
    B
    ` y(( tan ^(-1) x)/( x ) -(logx )/( 1+x^(2)))`
    C
    `( y)/(2) (( tan ^(-1) x)/( x ) +(logx )/( 1+x^(2)))`
    D
    `( y)/(2) (( tan ^(-1) x)/( x ) -(logx )/( 1+x^(2)))`

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