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Instead of the usual defination of deriv...

Instead of the usual defination of derivatie Df(x), if we define a new kind of derivatie `D^(**)F(x)` by the formula `D^(**)(x)=lim_(hrarr0) (f^(2)(x+h)-f^(2)(x))/(h)." where "f^(2)(x)" means "[f(x)]^(2)` and if f(x)=x log x, then `D^(**)f(x)|_(x=e)` has the value

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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