Given the function f, of, x, equals, minus, start fraction, 3, divided by, 4, end fraction, x, to the power 4 , plus, one fifth, x, to the power minus 4 , minus, one fifth, x, to the power 5*f*(*x*)=−43​*x*4+51​*x*−4−51​*x*5, find f, start superscript, prime, prime, end superscript, of, x*f*′′(*x*) in simplified form.

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.

Given the function f(x) = -3/4 x^4 + 1/5 x^-4 - 1/5 x^5, find f''(x) in simplified form.

Learn how to find the second derivative of the function f(x) = -3/4 x^4 + 1/5 x^-4 - 1/5 x^5 using the power rule. This step-by-step guide covers key calculus concepts, including finding first and second derivatives. Suitable for advanced high school or college students.

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