If f(0)=0 and f^{prime}(0)=3,then the derivat | vedclass

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(D) Given $f(0)=0$ and $f^{\prime}(0)=3$. Let $y = f(f(f(f(f(x)))))$. Using the chain rule,the derivative is: $\frac{dy}{dx} = f^{\prime}(f(f(f(f(x)))

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$243$

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(D) Given $f(0)=0$ and $f^{\prime}(0)=3$. Let $y = f(f(f(f(f(x)))))$. Using the chain rule,the derivative is: $\frac{dy}{dx} = f^{\prime}(f(f(f(f(x))))) \cdot f^{\prime}(f(f(f(x)))) \cdot f^{\prime}(f(f(x))) \cdot f^{\prime}(f(x)) \cdot f^{\prime}(x)$. At $x=0$: Since $f(0)=0$,we have $f(f(0)) = f(0) = 0$,and so on. Thus,$\left. \frac{dy}{dx} \right|_{x=0} = f^{\prime}(0) \cdot f^{\prime}(0) \cdot f^{\prime}(0) \cdot f^{\prime}(0) \cdot f^{\prime}(0) = [f^{\prime}(0)]^5$. Substituting $f^{\prime}(0)=3$: $[3]^5 = 243$.

If $f(0)=0$ and $f^{\prime}(0)=3$,then the derivative of $y=f(f(f(f(f(x)))))$ at $x=0$ is

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