To differentiate the expression \( \frac{2x^2 + 5x + 3}{x + 9} \) with respect to \( x \), we will use the **quotient rule**. The quotient rule states that if we have a function \( f(x) = \frac{u(x)}{v(x)} \), then its derivative is given by:

\[
f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]

Here, we have:
- \( u(x) = 2x^2 + 5x + 3 \)
- \( v(x) = x + 9 \)

Now, let's differentiate \( u(x) \) and \( v(x) \):
- \( u'(x) = 4x + 5 \)
- \( v'(x) = 1 \)

Using the quotient rule:

\[
f'(x) = \frac{(x + 9) \cdot (4x + 5) - (2x^2 + 5x + 3) \cdot 1}{(x + 9)^2}
\]

Now we expand and simplify:

\[
f'(x) = \frac{(x + 9)(4x + 5) - (2x^2 + 5x + 3)}{(x + 9)^2}
\]

Expanding both terms:

\[
(x + 9)(4x + 5) = 4x^2 + 5x + 36x + 45 = 4x^2 + 41x + 45
\]

Now subtract \( (2x^2 + 5x + 3) \):

\[
f'(x) = \frac{(4x^2 + 41x + 45) - (2x^2 + 5x + 3)}{(x + 9)^2}
\]

Simplifying the numerator:

\[
f'(x) = \frac{2x^2 + 36x + 42}{(x + 9)^2}
\]

So, the derivative of the function is:

\[
f'(x) = \frac{2x^2 + 36x + 42}{(x + 9)^2}
\]

Would you like more details or further clarifications?

Here are 5 related questions:
1. How is the quotient rule applied to more complex functions?
2. What other methods can be used to differentiate rational functions?
3. Can this differentiation be solved using implicit differentiation?
4. How do higher-order derivatives of rational functions behave?
5. What is the significance of critical points in such rational functions?

**Tip:** The quotient rule is particularly useful for differentiating rational functions where both the numerator and denominator involve \( x \). Always apply it carefully to avoid mistakes in expanding terms.

Learn how to differentiate the rational function (2x^2 + 5x + 3) / (x + 9) using the quotient rule. This detailed solution covers key calculus concepts such as differentiation, the quotient rule, and expanding expressions. Suitable for undergraduate students studying calculus.

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