Questions: Find (f^prime prime(x)). (f(x)=sqrt[4]left(x^2+1right)^3) (f^prime prime(x)=)

Find (f^prime prime(x)).
(f(x)=sqrt[4]left(x^2+1right)^3)
(f^prime prime(x)=)
Transcript text: Find $f^{\prime \prime}(x)$. \[ \begin{array}{l} f(x)=\sqrt[4]{\left(x^{2}+1\right)^{3}} \\ f^{\prime \prime}(x)=\square \end{array} \]
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Solution

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Solution Steps

To find the second derivative f(x) f^{\prime \prime}(x) of the function f(x)=(x2+1)34 f(x) = \sqrt[4]{(x^2 + 1)^3} , we first need to express the function in a form that is easier to differentiate. We can rewrite the function as f(x)=((x2+1)3)1/4 f(x) = ((x^2 + 1)^3)^{1/4} . Then, we apply the chain rule and power rule to find the first derivative f(x) f^{\prime}(x) . After obtaining the first derivative, we differentiate it again to find the second derivative f(x) f^{\prime \prime}(x) .

Step 1: Define the Function

We start with the function defined as: f(x)=(x2+1)34=((x2+1)3)14 f(x) = \sqrt[4]{(x^2 + 1)^3} = ((x^2 + 1)^3)^{\frac{1}{4}}

Step 2: Calculate the First Derivative

Using the chain rule and power rule, we find the first derivative f(x) f^{\prime}(x) : f(x)=34(x2+1)3412x=32x(x2+1)341(x2+1) f^{\prime}(x) = \frac{3}{4}(x^2 + 1)^{\frac{3}{4} - 1} \cdot 2x = \frac{3}{2} x (x^2 + 1)^{\frac{3}{4}} \cdot \frac{1}{(x^2 + 1)} This simplifies to: f(x)=3x2(x2+1)14 f^{\prime}(x) = \frac{3x}{2(x^2 + 1)^{\frac{1}{4}}}

Step 3: Calculate the Second Derivative

Next, we differentiate f(x) f^{\prime}(x) to find the second derivative f(x) f^{\prime \prime}(x) : f(x)=343x2(x2+1)54+321(x2+1)14 f^{\prime \prime}(x) = -\frac{3}{4} \cdot \frac{3x^2}{(x^2 + 1)^{\frac{5}{4}}} + \frac{3}{2} \cdot \frac{1}{(x^2 + 1)^{\frac{1}{4}}} This can be simplified to: f(x)=(0.75x2+1.5)(x2+1)34(x2+1)2 f^{\prime \prime}(x) = \frac{(0.75x^2 + 1.5)(x^2 + 1)^{\frac{3}{4}}}{(x^2 + 1)^2}

Final Answer

Thus, the second derivative is: f(x)=(0.75x2+1.5)(x2+1)34(x2+1)2 \boxed{f^{\prime \prime}(x) = \frac{(0.75x^2 + 1.5)(x^2 + 1)^{\frac{3}{4}}}{(x^2 + 1)^2}}

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