differentiate with respect to x 3x+3/ 2x+3

To differentiate the function \( f(x) = \frac{3x + 3}{2x + 3} \) with respect to \( x \), we will use the **quotient rule**. The quotient rule is given by:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]

Here, \( u(x) = 3x + 3 \) and \( v(x) = 2x + 3 \). Now, let's differentiate both \( u(x) \) and \( v(x) \):

- \( u'(x) = 3 \)
- \( v'(x) = 2 \)

Now, apply the quotient rule:

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

Simplifying the numerator:

\[
f'(x) = \frac{6x + 9 - (6x + 6)}{(2x + 3)^2}
\]

\[
f'(x) = \frac{6x + 9 - 6x - 6}{(2x + 3)^2}
\]

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

Thus, the derivative of \( \frac{3x + 3}{2x + 3} \) is:

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

Would you like more details on this or have any other questions?

Here are 5 related questions:
1. How do you apply the product rule to differentiate functions?
2. What happens when you differentiate a function involving a square root?
3. How do you find the second derivative of a quotient?
4. How is implicit differentiation different from explicit differentiation?
5. How do you differentiate trigonometric functions?

**Tip**: Always simplify the function as much as possible before applying differentiation rules.

Learn how to differentiate the function (3x + 3) / (2x + 3) step-by-step using the quotient rule. This solution includes key calculus concepts such as differentiation and provides a detailed breakdown of the formula used. Suitable for students in Grades 11-12.

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