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Question
Differentiate the following w.r.t. x: `x^(tan^(-1)x`
Solution
Let y = `x^(tan^(-1)x`
Then log y = `log (x^(tan^(-1)x)) = (tan^-1 x)(logx)`
Differentiating both sides w.r.t. x, we get
`(1)/y.(dy)/(dx) = d/(dx)[(tan^-1 x)(logx)]`
= `(tan^-1 x).d/(dx)(logx) + (logx).d/(dx)(tan^-1 x)`
= `(tan^-1 x) xx 1/x + (logx) xx 1/(1 + x^2)`
∴ `(dy)/(dx) = y[(tan^-1 x)/x + logx/(1 + x^2)]`
= `x^(tan^-1 x)[tan^-1 x/x + logx/(1 + x^2)]`
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