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The problem asks for the derivative of the function \((x + 2)(5x + 9)\) using the Product Rule. The Product Rule states that if you have two functions \(u(x)\) and \(v(x)\), then the derivative of their product is given by:

\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x).
\]

Let:
- \( u(x) = x + 2 \) and \( v(x) = 5x + 9 \).

Now, find the derivatives of \( u(x) \) and \( v(x) \):
- \( u'(x) = 1 \) (since the derivative of \(x + 2\) is 1).
- \( v'(x) = 5 \) (since the derivative of \(5x + 9\) is 5).

Apply the Product Rule:
\[
\frac{d}{dx}[(x + 2)(5x + 9)] = u'(x)v(x) + u(x)v'(x) = (1)(5x + 9) + (x + 2)(5).
\]

Now, simplify:
\[
5x + 9 + 5(x + 2) = 5x + 9 + 5x + 10.
\]

Combine like terms:
\[
10x + 19.
\]

The derivative of \((x + 2)(5x + 9)\) is indeed \(10x + 19\), which matches the answer provided.

Would you like more details or have any other questions?

### Follow-Up Questions:
1. How would the result change if one of the functions had a higher degree?
2. Can you explain why the Product Rule is necessary for derivatives of products?
3. What would be the derivative of \((x^2 + 2)(5x + 9)\) using the Product Rule?
4. How does the Product Rule differ from the Quotient Rule?
5. Why is it important to simplify derivatives after applying differentiation rules?

### Tip:
When using the Product Rule, always identify your \(u(x)\) and \(v(x)\) clearly, and calculate their derivatives separately to avoid mistakes.

Use the Product Rule to find the derivative of (x + 2)(5x + 9).

Learn how to apply the Product Rule to find the derivative of the function (x + 2)(5x + 9). This solution covers key calculus concepts and provides a step-by-step simplification process. Suitable for high school or early university calculus students.

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