We are given the function:

$f(x) = \sqrt[5]{(x^2 + 1)^4}$

We need to find $f''(x)$, the second derivative of $f(x)$.

Step 1: Rewrite the function

First, let's express the function in a more convenient form for differentiation. The fifth root can be rewritten as a power:

$f(x) = \left( (x^2 + 1)^4 \right)^{\frac{1}{5}} = (x^2 + 1)^{\frac{4}{5}}$

Step 2: First derivative $f'(x)$

To find the first derivative $f'(x)$, we will use the chain rule. The outer function is $u^{\frac{4}{5}}$, and the inner function is $u = x^2 + 1$.

1. Differentiate the outer function:

$\frac{d}{du} \left( u^{\frac{4}{5}} \right) = \frac{4}{5} u^{-\frac{1}{5}}$

2. Now, differentiate the inner function $u = x^2 + 1$:

$\frac{d}{dx} (x^2 + 1) = 2x$

Using the chain rule, the first derivative is:

$f'(x) = \frac{4}{5} (x^2 + 1)^{-\frac{1}{5}} \cdot 2x = \frac{8x}{5} (x^2 + 1)^{-\frac{1}{5}}$

Step 3: Second derivative $f''(x)$

Now, to find the second derivative $f''(x)$, we need to apply the product rule and the chain rule to the first derivative:

$f'(x) = \frac{8x}{5} (x^2 + 1)^{-\frac{1}{5}}$

We will differentiate this expression using the product rule, where the two factors are $\frac{8x}{5}$ and $(x^2 + 1)^{-\frac{1}{5}}$.

1. Differentiate $\frac{8x}{5}$:

$\frac{d}{dx} \left( \frac{8x}{5} \right) = \frac{8}{5}$

2. Differentiate $(x^2 + 1)^{-\frac{1}{5}}$:

Using the chain rule:

$\frac{d}{dx} \left( (x^2 + 1)^{-\frac{1}{5}} \right) = -\frac{1}{5} (x^2 + 1)^{-\frac{6}{5}} \cdot 2x = -\frac{2x}{5} (x^2 + 1)^{-\frac{6}{5}}$

Now, applying the product rule:

$f''(x) = \frac{8}{5} (x^2 + 1)^{-\frac{1}{5}} + \frac{8x}{5} \cdot \left( -\frac{2x}{5} (x^2 + 1)^{-\frac{6}{5}} \right)$

Simplify the second term:

$f''(x) = \frac{8}{5} (x^2 + 1)^{-\frac{1}{5}} - \frac{16x^2}{25} (x^2 + 1)^{-\frac{6}{5}}$

Final answer:

$f''(x) = \frac{8}{5} (x^2 + 1)^{-\frac{1}{5}} - \frac{16x^2}{25} (x^2 + 1)^{-\frac{6}{5}}$

Would you like to explore any steps further or have specific questions?

5 Related Questions:

1. How does the chain rule apply to nested functions like $(x^2 + 1)^4$?
2. Can the second derivative be used to find concavity of this function?
3. How would the derivative change if the exponent of $x^2 + 1$ were different?
4. What is the importance of the product rule in differentiating composite functions?
5. How can higher-order derivatives help in approximating functions?

Tip:

When applying the product rule, carefully differentiate each factor while keeping the chain rule in mind for composite expressions.