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The function given is:

\[
f(x) = \frac{2}{x} + 3x^2 - 3, \quad x \neq 0.
\]

To find \( f'(x) \), we need to differentiate each term of \( f(x) \) with respect to \( x \).

### Step-by-Step Solution

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

2. **Differentiate** \( 3x^2 \):
   \[
   \frac{d}{dx} (3x^2) = 6x.
   \]

3. **Differentiate** the constant term \( -3 \):
   \[
   \frac{d}{dx} (-3) = 0.
   \]

### Combine Results

Now, we combine these results to obtain \( f'(x) \):
\[
f'(x) = -\frac{2}{x^2} + 6x.
\]

### Final Answer

\[
f'(x) = -\frac{2}{x^2} + 6x.
\]

Would you like further details or have any questions?

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**Additional questions for deeper understanding:**

1. How would \( f'(x) \) change if \( f(x) \) included a term with \( x^3 \)?
2. What does \( f'(x) = 0 \) imply about the function \( f(x) \) at certain points?
3. How would the derivative change if the constant term \( -3 \) were different?
4. Could we interpret \( f'(x) \) geometrically? What does it tell us about the slope of \( f(x) \)?
5. How does the differentiation of \( \frac{2}{x} \) change if we consider higher powers, like \( \frac{2}{x^n} \)?

**Tip:** Always rewrite terms with negative exponents or fractions before differentiating; it simplifies the process.

The function f is defined by f(x) = 2/x + 3x^2 - 3, x ≠ 0. (a) Find f'(x).

Learn how to find the derivative of f(x) = 2/x + 3x^2 - 3 with a step-by-step solution. This calculus problem covers differentiation techniques, including the power rule and derivatives of rational functions. Suitable for advanced high school students in Grades 11-12.

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