To differentiate $(2x - 7)^3 \cdot (x^2 - 3x - 2)^3$, we will apply the product rule and chain rule.

Step-by-Step Solution:

Let: $u = (2x - 7)^3 \quad \text{and} \quad v = (x^2 - 3x - 2)^3$

The product rule for derivatives is: $\frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v'$ Thus, we need to find $u'$ and $v'$.

1. Differentiate $u = (2x - 7)^3$:

Using the chain rule: $u' = 3(2x - 7)^2 \cdot \frac{d}{dx}(2x - 7) = 3(2x - 7)^2 \cdot 2 = 6(2x - 7)^2$

2. Differentiate $v = (x^2 - 3x - 2)^3$:

Again using the chain rule: $v' = 3(x^2 - 3x - 2)^2 \cdot \frac{d}{dx}(x^2 - 3x - 2) = 3(x^2 - 3x - 2)^2 \cdot (2x - 3)$

3. Apply the product rule:

Now, substitute these results into the product rule: $\frac{d}{dx} \left[ (2x - 7)^3 \cdot (x^2 - 3x - 2)^3 \right] = u'v + uv'$ This becomes: $= 6(2x - 7)^2 \cdot (x^2 - 3x - 2)^3 + (2x - 7)^3 \cdot 3(x^2 - 3x - 2)^2 \cdot (2x - 3)$

Final Answer:

$\frac{d}{dx} \left[ (2x - 7)^3 \cdot (x^2 - 3x - 2)^3 \right] = 6(2x - 7)^2 (x^2 - 3x - 2)^3 + 3(2x - 7)^3 (x^2 - 3x - 2)^2 (2x - 3)$

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

Relative Questions:

1. How does the chain rule work in differentiation?
2. What is the product rule, and how is it applied?
3. Can you explain the difference between $u'$ and $v'$ in the product rule?
4. How do you simplify a derivative expression?
5. What happens if one of the functions in the product rule is constant?

Tip: In any derivative involving products or compositions of functions, it’s essential to carefully apply the product rule and chain rule to avoid errors.