If f is a polynomial,then the second derivati | vedclass

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

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(D) Let $y = f(e^x)$.First,we find the first derivative with respect to $x$ using the chain rule:$\frac{dy}{dx} = f'(e^x) \cdot \frac{d}{dx}(e^x) = f'(e^x)e^x$.Now,we find the second derivative by applying the product rule to $\frac{dy}{dx} = f'(e^x)e^x$:$\frac{d^2y}{dx^2} = \frac{d}{dx}(f'(e^x)e^x) = \frac{d}{dx}(f'(e^x)) \cdot e^x + f'(e^x) \cdot \frac{d}{dx}(e^x)$.Using the chain rule again for $\frac{d}{dx}(f'(e^x)) = f''(e^x)e^x$,we get:$\frac{d^2y}{dx^2} = (f''(e^x)e^x)e^x + f'(e^x)e^x$.Simplifying this,we obtain:$\frac{d^2y}{dx^2} = f''(e^x)e^{2x} + f'(e^x)e^x$.

If $f$ is a polynomial,then the second derivative of $f(e^x)$ is

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