To find the second derivative of the function

\[
f(x) = -\frac{3}{x^2} + 3
\]

let's proceed with the following steps:

1. **Rewrite the Function:** Rewrite \(-\frac{3}{x^2}\) using negative exponents:
   \[
   f(x) = -3x^{-2} + 3
   \]

2. **First Derivative:** Use the power rule \(\frac{d}{dx}(x^n) = n \cdot x^{n-1}\):
   \[
   f'(x) = -3 \cdot (-2) \cdot x^{-3} = 6x^{-3}
   \]
   This simplifies to:
   \[
   f'(x) = \frac{6}{x^3}
   \]

3. **Second Derivative:** Differentiate \(f'(x) = 6x^{-3}\) again using the power rule:
   \[
   f''(x) = 6 \cdot (-3) \cdot x^{-4} = -18x^{-4}
   \]
   This simplifies to:
   \[
   f''(x) = -\frac{18}{x^4}
   \]

So, the second derivative is:

\[
f''(x) = -\frac{18}{x^4}
\]

Would you like details on any specific part of the process?

---

1. What is the general power rule for derivatives?
2. How would the derivatives change if the function was \(-\frac{3}{x^3} + 3\)?
3. What are higher-order derivatives, and how are they used?
4. How does the sign of the second derivative relate to concavity?
5. How does changing the constant in the function affect the derivatives?

**Tip:** When working with negative exponents, rewrite them as fractions if it helps simplify calculations.

What's the second derivative of -3/x^2 + 3?

Discover how to find the second derivative of the function -3/x^2 + 3, using calculus concepts such as the power rule and negative exponents. This solution includes a step-by-step breakdown and explains each calculation in detail. Ideal for students in Grades 11-12.

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