1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

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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 $D^* f(x)=\underset{h \rightarrow 0}{\operatorname{Limit}} \frac{f^2(x+h)-f^2(x)}{h}$ where $f^2(x)$ means $[f(x)]^2$. If $f(x)=x \ln x$ then $\left.D * f(x)\right|_{x=e}$ has the value

Continuity and Differentiability

Solution:

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