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Instead of the usual definition of derivative `Df(x),` if we define a new kind of derivative `D^*F(x)` by the formula `D*(x)=(lim)_(hvec0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x)` mean `[f(x)]^2a n diff(x)=xlogx ,t h e nD^*f(x)(""|)_(x=e)` has the value e (B) 2e (c) 4e (d) none of these

A

e

B

2e

C

4e

D

none of these

Text Solution

Verified by Experts

`D^(**)(x)=underset(hrarr0)lim(f^(2)(x+h)-f^(2)(x))/(h)`
`=underset(hrarr0)lim(f(x+h)-f(x))/(h)(f(x+_h)+f(x))`
`=2f(x)underset(hrarr0)lim(f(x+h)-f(x))/(h)`
`=2f(x)cdotf'(x)`
`"or "D^(**)(x log x)=2x log x (1+ log x)`
`"or "D^(**)f(x):|_(x=e)=4e`
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