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If f(x)=(log)(x^2)(logx), then f^(prime)...

If `f(x)=(log)_(x^2)(logx),` then `f^(prime)(x)` at `x=e` is 0 (b) 1 (c) `1/e` (d) `1/2e`

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`text { Given, } f(x)=log _{x^{2}}(log _{e} x)=frac{1}{2} log _{x}(log _{e} x)`

`Rightarrow f(x)=frac{1}{2} frac{log _{e} log _{e} x}{log _{e} x}`

On differentiating both sides with respect to `{x}`, we get

`f^{prime}(x)=frac{1}{2} frac{log _{e} x(frac{1}{x log _{e} x})-log _{e} log _{e} x times frac{1}{x}}{(log _{e} x)^{2}}`

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