Find the derivative of the function.
f(x) = 
x^3 + 9x + 8/x^2 − 7


We need to find the derivative of the function \( f(x) = x^3 + 9x + \frac{8}{x^2} - 7 \).

### Step-by-step differentiation:

1. The derivative of \( x^3 \) is:
   \[
   \frac{d}{dx} \left( x^3 \right) = 3x^2
   \]

2. The derivative of \( 9x \) is:
   \[
   \frac{d}{dx} \left( 9x \right) = 9
   \]

3. For \( \frac{8}{x^2} \), we can rewrite it as \( 8x^{-2} \) and then differentiate:
   \[
   \frac{d}{dx} \left( 8x^{-2} \right) = 8 \cdot (-2) x^{-3} = -\frac{16}{x^3}
   \]

4. The derivative of the constant \( -7 \) is:
   \[
   \frac{d}{dx} (-7) = 0
   \]

### Now, summing up the derivatives:
\[
f'(x) = 3x^2 + 9 - \frac{16}{x^3}
\]

This is the derivative of the function.

Do you want details or have any questions?

### Related Questions:
1. What is the second derivative of \( f(x) \)?
2. How do we find the critical points of \( f(x) \)?
3. Can you explain the chain rule for composite functions?
4. What is the derivative of \( \ln(x) \)?
5. How does the power rule apply to negative exponents?

**Tip**: Always simplify the expression before differentiating—it can make the process easier!

Find the derivative of the function f(x) = x^3 + 9x + 8/x^2 − 7.

Learn how to differentiate the function f(x) = x^3 + 9x + 8/x^2 − 7 using the power rule and negative exponents. This detailed step-by-step solution is suitable for advanced high school or college-level calculus students.

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