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

Solution Steps

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

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

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

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

Final Answer