Questions: Find (f''(x)). (f(x)=(x^2+4)^6) (f''(x)=)

Find (f''(x)). (f(x)=(x^2+4)^6) (f''(x)=)
Transcript text: Find $f^{\prime \prime}(x)$. \[ f(x)=\left(x^{2}+4\right)^{6} \] \[ f^{\prime \prime}(x)= \]
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Solution

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Solution Steps

Step 1: Find the First Derivative

To find the first derivative \( f^{\prime}(x) \) of the function \( f(x) = (x^2 + 4)^6 \), we apply the chain rule. The result is: \[ f^{\prime}(x) = 12x(x^2 + 4)^5 \]

Step 2: Find the Second Derivative

Next, we differentiate the first derivative \( f^{\prime}(x) \) to obtain the second derivative \( f^{\prime \prime}(x) \). The calculation yields: \[ f^{\prime \prime}(x) = 120x^2(x^2 + 4)^4 + 12(x^2 + 4)^5 \]

Final Answer

Thus, the second derivative of the function is: \[ \boxed{f^{\prime \prime}(x) = 120x^2(x^2 + 4)^4 + 12(x^2 + 4)^5} \]

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