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underset(hrarr0)lim((e+h)^(In(e+h))-e)/(...

`underset(hrarr0)lim((e+h)^(In(e+h))-e)/(h)` is-

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`"Limit is "f'(e)" where "f(x)=x^(In x)=e^(In^(2)x)`
`therefore" "f'(x)=e^(In^(2)x).(2 In x)/(x)`
`"or "f'(e)=e.(2)/(e)=2`
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Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h) =lim_(hrarr0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h) =lim_(hrarr0^(+))(f(a)-f(a+h))/(h) =lim_(hrarr0^(+)) (f(a)-f(x))/(a-x) respectively. Let f be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function. If f is even function, which of the following is right hand derivative of f' at x=a?

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