To find the derivative of $f(x) = (x^3 - x)(x^2 + 2)(x + 2)$, we will apply the product rule. The product rule states:

$(fg)' = f'g + fg'$

For three functions, say $u(x), v(x), w(x)$, the product rule extends as:

$(uvw)' = u'vw + uv'w + uvw'$

Here, we have: $u(x) = x^3 - x,\quad v(x) = x^2 + 2,\quad w(x) = x + 2$

Now, we will find the derivatives of each:

1. $u'(x) = 3x^2 - 1$
2. $v'(x) = 2x$
3. $w'(x) = 1$

Now, apply the extended product rule:

$f'(x) = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)$

Substitute the functions and their derivatives:

$f'(x) = (3x^2 - 1)(x^2 + 2)(x + 2) + (x^3 - x)(2x)(x + 2) + (x^3 - x)(x^2 + 2)(1)$

This is the derivative of $f(x)$, but we can simplify the expression if needed.

Would you like me to simplify this derivative further, or do you have any questions about the steps?

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5 Related Questions:

1. How do you apply the product rule for more than two functions?
2. What is the chain rule, and how is it different from the product rule?
3. How can you simplify the result of a derivative expression like this one?
4. What is the derivative of a polynomial function like $x^3 - x$?
5. Can you explain when to use the quotient rule versus the product rule?

Tip:

When differentiating products of multiple functions, always check if any factors can be simplified before applying the product rule.