MathematicsFor a differentiable function, define D∗f(x)=limh→0⁡f2(x+h)−f2(x)h where f2(x)=(f(x))2 for example, D∗f(x)=2x if f(x)=x, D∗f(x)=2sin⁡xcos⁡x if f(x)=sin⁡xD∗f(x)=2e2x if f(x)=ex

For a differentiable function, define D∗f(x)=limh→0⁡f2(x+h)−f2(x)h where f2(x)=(f(x))2 for example, D∗f(x)=2x if f(x)=x, D∗f(x)=2sin⁡xcos⁡x if f(x)=sin⁡xD∗f(x)=2e2x if f(x)=ex

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    Solution:

    For a differentiable function, define \( D∗f(x) = \lim_{h \to 0} \frac{f^2(x+h) - f^2(x)}{h} \) where \( f^2(x) = (f(x))^2 \).

    For example:

    • \( D∗f(x) = 2x \) if \( f(x) = x \)
    • \( D∗f(x) = 2 \sin(x) \cos(x) \) if \( f(x) = \sin(x) \)
    • \( D∗f(x) = 2 e^{2x} \) if \( f(x) = e^x \)

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