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

To differentiate the expression \(\frac{3x + 3}{2x + 3}\) with respect to \(x\), we will apply the **quotient rule** of differentiation. The quotient rule states that for a function \(\frac{u(x)}{v(x)}\), the derivative 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 \(v(x) = 2x + 3\).

### Step 1: Differentiate \(u(x)\) and \(v(x)\)
- \(u'(x) = \frac{d}{dx}(3x + 3) = 3\)
- \(v'(x) = \frac{d}{dx}(2x + 3) = 2\)

### Step 2: Apply the quotient rule
\[
\frac{d}{dx} \left( \frac{3x + 3}{2x + 3} \right) = \frac{(3)(2x + 3) - (3x + 3)(2)}{(2x + 3)^2}
\]

### Step 3: Simplify the numerator
\[
= \frac{6x + 9 - (6x + 6)}{(2x + 3)^2}
\]
\[
= \frac{6x + 9 - 6x - 6}{(2x + 3)^2}
\]
\[
= \frac{3}{(2x + 3)^2}
\]

### Final Answer:
\[
\frac{d}{dx} \left( \frac{3x + 3}{2x + 3} \right) = \frac{3}{(2x + 3)^2}
\]

Would you like more details or have any further questions?

Here are five related questions to expand your understanding:

1. How do you differentiate a product of two functions?
2. What is the chain rule and how does it apply to composite functions?
3. How would the quotient rule change if \(u(x)\) or \(v(x)\) were more complex functions?
4. Can you differentiate \( \frac{5x + 2}{3x^2 + 1} \) using the quotient rule?
5. What are the conditions under which the quotient rule can be applied?

**Tip:** When using the quotient rule, always double-check the sign and terms in the numerator after differentiating each part.

Learn how to differentiate the rational function (3x+3)/(2x+3) using the quotient rule. This solution breaks down the steps, including applying the quotient rule, simplifying the expression, and finding the derivative. Ideal for students in Grades 11-12 studying calculus.

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