To find the derivative of the function $(x^2-4)(x-2)$ using the product rule, we'll follow these steps:

1. Product Rule

The product rule states that if we have a function $f(x) = g(x) \cdot h(x)$, then its derivative is given by: $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$

2. Define $g(x)$ and $h(x)$

Let: $g(x) = x^2 - 4$ $h(x) = x - 2$

3. Find $g'(x)$ and $h'(x)$

Compute the derivatives of $g(x)$ and $h(x)$.

• For $g(x) = x^2 - 4$: $g'(x) = 2x$

• For $h(x) = x - 2$: $h'(x) = 1$

4. Apply the Product Rule

Using the product rule, we have: $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$ Substitute $g'(x)$, $g(x)$, $h'(x)$, and $h(x)$ into the formula:

$f'(x) = (2x) \cdot (x - 2) + (x^2 - 4) \cdot (1)$

5. Simplify the Expression

Now, simplify each term:

• $(2x) \cdot (x - 2) = 2x^2 - 4x$
• $(x^2 - 4) \cdot (1) = x^2 - 4$

So, the derivative is:

$f'(x) = 2x^2 - 4x + x^2 - 4$

Combine like terms:

$f'(x) = 3x^2 - 4x - 4$

Final Answer

The derivative of the function $(x^2-4)(x-2)$ is:

$f'(x) = 3x^2 - 4x - 4$

Would you like more details or have any other questions?

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Related Questions:

1. How would you find the derivative of $(x^2 - 4)(x + 3)$ using the product rule?
2. What happens if you apply the quotient rule to $\frac{x^2 - 4}{x - 2}$?
3. Can you find the second derivative of $(x^2-4)(x-2)$?
4. How would you find the critical points of the function $f(x) = (x^2-4)(x-2)$?
5. What is the derivative of $(x^2-4)^2$ using the chain rule?

Tip: When using the product rule, always carefully identify and differentiate both functions involved before substituting into the rule.