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(D) Let $y = f(e^x)$.First,we find the first derivative with respect to $x$ using the chain rule:$\frac{dy}{dx} = f'(e^x) \cdot \frac{d}{dx}(e^x) = f'

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$f''(e^x) \cdot e^{2x} + f'(e^x) \cdot e^x$

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(D) Let $y = f(e^x)$.First,we find the first derivative with respect to $x$ using the chain rule:$\frac{dy}{dx} = f'(e^x) \cdot \frac{d}{dx}(e^x) = f'(e^x) \cdot e^x$.Now,we find the second derivative by differentiating $\frac{dy}{dx}$ with respect to $x$ using the product rule:$\frac{d^2y}{dx^2} = \frac{d}{dx}[f'(e^x) \cdot e^x]$$= \frac{d}{dx}[f'(e^x)] \cdot e^x + f'(e^x) \cdot \frac{d}{dx}(e^x)$$= [f''(e^x) \cdot e^x] \cdot e^x + f'(e^x) \cdot e^x$$= f''(e^x) \cdot e^{2x} + f'(e^x) \cdot e^x$.Thus,the correct option is $D$.

Let $f(x)$ be a polynomial in $x$. Then the second derivative of $f(e^x)$ is:

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