Find the derivative of f(x)=
3x3
x2–4
.
f′(x)=



To find the derivative of the function \( f(x) = \frac{3x^3}{x^2 - 4} \), we can use the **quotient rule**. The quotient rule states that if we have a function \( f(x) = \frac{g(x)}{h(x)} \), then

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]

where \( g(x) = 3x^3 \) and \( h(x) = x^2 - 4 \).

### Step 1: Differentiate \( g(x) \) and \( h(x) \)
1. \( g(x) = 3x^3 \)  
   \( g'(x) = 9x^2 \)

2. \( h(x) = x^2 - 4 \)  
   \( h'(x) = 2x \)

### Step 2: Apply the Quotient Rule
Now, using the quotient rule:

\[
f'(x) = \frac{(9x^2)(x^2 - 4) - (3x^3)(2x)}{(x^2 - 4)^2}
\]

### Step 3: Simplify the Expression
Let's simplify the terms in the numerator:

1. Expanding \( (9x^2)(x^2 - 4) \):
   \[
   9x^2 \cdot x^2 - 9x^2 \cdot 4 = 9x^4 - 36x^2
   \]

2. Expanding \( (3x^3)(2x) \):
   \[
   3x^3 \cdot 2x = 6x^4
   \]

So, the numerator becomes:

\[
9x^4 - 36x^2 - 6x^4 = 3x^4 - 36x^2
\]

Therefore, we have:

\[
f'(x) = \frac{3x^4 - 36x^2}{(x^2 - 4)^2}
\]

### Final Answer
\[
f'(x) = \frac{3x^2(x^2 - 12)}{(x^2 - 4)^2}
\]

Would you like further clarification on any steps?

---

1. How is the quotient rule applied to complex functions?
2. What happens to \( f'(x) \) if \( x = 2 \) or \( x = -2 \)?
3. Could the function be simplified before differentiation?
4. What is the importance of simplification in calculus?
5. How do we handle higher-order derivatives of rational functions?

**Tip:** Simplifying expressions before applying differentiation rules can sometimes make calculations much easier.

Find the derivative of f(x) = 3x^3 / (x^2 - 4).

Learn how to find the derivative of f(x) = 3x^3 / (x^2 - 4) using the quotient rule. This solution includes a step-by-step application of differentiation techniques, simplifying complex fractions, and essential calculus concepts. Ideal for Grade 12 and college-level calculus students.

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