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(C) Given the definition $D^*f(x) = \lim_{h 	o 0} \frac{f^2(x + h) - f^2(x)}{h}$.This is the definition of the derivative of the function $g(x) = f^2

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$4e$

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(C) Given the definition $D^*f(x) = \lim_{h 	o 0} \frac{f^2(x + h) - f^2(x)}{h}$.This is the definition of the derivative of the function $g(x) = f^2(x) = [f(x)]^2$.Using the chain rule,$D^*f(x) = \frac{d}{dx} [f(x)]^2 = 2f(x) \cdot f'(x)$.Given $f(x) = x \ln x$,we find $f'(x)$ using the product rule:$f'(x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1$.Now,substitute $f(x)$ and $f'(x)$ into the expression for $D^*f(x)$:$D^*f(x) = 2(x \ln x)(\ln x + 1)$.Evaluate at $x = e$:$D^*f(e) = 2(e \ln e)(\ln e + 1) = 2(e \cdot 1)(1 + 1) = 2e(2) = 4e$.

People living on Mars,instead of the usual definition of derivative $D f(x)$,define a new kind of derivative,$D^f(x)$ by the formula $D^f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}$ where $f^2(x)$ means $[f(x)]^2$. If $f(x) = x \ln x$,then the value of $\left. D^*f(x) \right|_{x = e}$ is:

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People living on Mars,instead of the usual definition of derivative $D f(x)$,define a new kind of derivative,$D^*f(x)$ by the formula $D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}$ where $f^2(x)$ means $[f(x)]^2$. If $f(x) = x \ln x$,then the value of $\left. D^*f(x) \right|_{x = e}$ is: