Find the derivative of the function.
f(x) = 	x^3 − 7x^2 + 1/
x^2


To find the derivative of the function \( f(x) = x^3 - 7x^2 + \frac{1}{x^2} \), we will differentiate each term separately. Let's go through this step-by-step.

### Step 1: Rewrite the Function
First, rewrite the function to simplify the differentiation process:
\[
f(x) = x^3 - 7x^2 + \frac{1}{x^2}
\]
The term \(\frac{1}{x^2}\) can be rewritten as \(x^{-2}\). So the function becomes:
\[
f(x) = x^3 - 7x^2 + x^{-2}
\]

### Step 2: Differentiate Each Term
Now, apply the power rule of differentiation, which states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

1. **Differentiate \(x^3\):**
   \[
   \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2
   \]

2. **Differentiate \(-7x^2\):**
   \[
   \frac{d}{dx}(-7x^2) = -7 \cdot 2x^{2-1} = -14x
   \]

3. **Differentiate \(x^{-2}\):**
   \[
   \frac{d}{dx}(x^{-2}) = -2x^{-2-1} = -2x^{-3}
   \]

### Step 3: Combine the Derivatives
Combine the derivatives of each term to get the derivative of the entire function:
\[
f'(x) = 3x^2 - 14x - 2x^{-3}
\]

### Step 4: Rewrite in Standard Form
You can rewrite \( -2x^{-3} \) as \(-\frac{2}{x^3}\) to present the derivative in a more standard form:
\[
f'(x) = 3x^2 - 14x - \frac{2}{x^3}
\]

Thus, the derivative of the function \( f(x) = x^3 - 7x^2 + \frac{1}{x^2} \) is:
\[
f'(x) = 3x^2 - 14x - \frac{2}{x^3}
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the second derivative of \(f(x) = x^3 - 7x^2 + \frac{1}{x^2}\)?
2. What is the derivative of \(f(x) = x^3 + x^{-2}\)?
3. How do you apply the power rule to negative exponents?
4. What are the critical points of the function \(f(x) = x^3 - 7x^2 + \frac{1}{x^2}\)?
5. How does the graph of \(f(x) = x^3 - 7x^2 + \frac{1}{x^2}\) look compared to its derivative?

**Tip:** When differentiating functions with negative exponents, remember to decrease the exponent by one and multiply by the original exponent.

Learn how to find the derivative of the function f(x) = x^3 - 7x^2 + 1/x^2 step-by-step. This problem covers differentiation techniques, including the power rule and handling negative exponents. Suitable for college-level mathematics.

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