To find the second derivative of the function $f(x) = -\frac{3}{4} x^4 + \frac{1}{5} x^{-4} - \frac{1}{5} x^5$, we need to follow these steps:

Step 1: Find the first derivative, $f'(x)$

Using the power rule, which states that the derivative of $x^n$ is $n x^{n-1}$, we differentiate each term:

1. $-\frac{3}{4} x^4$ becomes $-\frac{3}{4} \times 4 x^{4-1} = -3x^3$
2. $\frac{1}{5} x^{-4}$ becomes $\frac{1}{5} \times (-4) x^{-4-1} = -\frac{4}{5} x^{-5}$
3. $-\frac{1}{5} x^5$ becomes $-\frac{1}{5} \times 5 x^{5-1} = -x^4$

So, the first derivative is: $f'(x) = -3x^3 - \frac{4}{5} x^{-5} - x^4$

Step 2: Find the second derivative, $f''(x)$

Now, differentiate $f'(x)$ term by term:

1. $-3x^3$ becomes $-3 \times 3 x^{3-1} = -9x^2$
2. $-\frac{4}{5} x^{-5}$ becomes $-\frac{4}{5} \times (-5) x^{-5-1} = 4x^{-6}$
3. $-x^4$ becomes $-4x^{4-1} = -4x^3$

So, the second derivative is: $f''(x) = -9x^2 + 4x^{-6} - 4x^3$

Final Answer:

$f''(x) = -9x^2 + 4x^{-6} - 4x^3$

Would you like more details or have any questions?

Here are five related questions you might explore:

1. What is the third derivative of the given function?
2. What is the behavior of $f(x)$ as $x \to \infty$?
3. How does the graph of $f(x)$ change with respect to its second derivative?
4. Can you find the critical points of $f(x)$?
5. What is the concavity of $f(x)$ based on the second derivative?

Tip: When taking higher-order derivatives, carefully apply the power rule for each term independently to avoid mistakes.