People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D * f ( x ) by the formula D* f(x)= underseth arrow 0 operatornameLimit (f2(x+h)-f2(x)/h) where f2(x) means [f(x)]2. If f(x)=x ln x then .D * f(x)|x=e has the value

Question

People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D * f ( x ) by the formula D* f(x)= underseth arrow 0 operatornameLimit (f2(x+h)-f2(x)/h) where f2(x) means [f(x)]2. If f(x)=x ln x then .D * f(x)|x=e has the value (A) e (B) 2 e (C) 4 e (D) 8 e

Tardigrade Admin · Accepted Answer

D * f ( x )=2 f ( x ) ⋅ f prime( x ) D *( x ln x )=2 x ln x (1+ ln x )

$e$

$2 e$

$4 e$

$8 e$

$D^{} f (x) = 2 f (x) \cdot f^{'} (x)$
$D^{} (x ln x) = 2 x ln x (1 + ln x)$

Q. People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D∗f(x) by the formula D∗f(x)=h→0Limit​hf2(x+h)−f2(x)​ where f2(x) means [f(x)]2. If f(x)=xlnx then D∗f(x)∣x=e​ has the value

e

2e

4e

8e

D∗f(x)=2f(x)⋅f′(x) D∗(xlnx)=2xlnx(1+lnx)

$e$

$2 e$

$4 e$

$8 e$

$D^{} f (x) = 2 f (x) \cdot f^{'} (x)$
$D^{} (x ln x) = 2 x ln x (1 + ln x)$