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

To differentiate the expression \(\frac{3x + 2}{2x + 3}\) with respect to \(x\), we will apply the **quotient rule**, which states:

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

Let:
- \(f(x) = 3x + 2\)
- \(g(x) = 2x + 3\)

Now, let's compute the derivatives:
- \(f'(x) = 3\) (since the derivative of \(3x + 2\) is 3)
- \(g'(x) = 2\) (since the derivative of \(2x + 3\) is 2)

Using the quotient rule:

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

Now, let's simplify the numerator:

\[
\text{Numerator} = 3(2x + 3) - 2(3x + 2) = (6x + 9) - (6x + 4) = 6x + 9 - 6x - 4 = 5
\]

Thus, the derivative is:

\[
\frac{5}{(2x + 3)^2}
\]

### Final Answer:
\[
\frac{d}{dx}\left(\frac{3x + 2}{2x + 3}\right) = \frac{5}{(2x + 3)^2}
\]

Would you like more details on any of the steps?

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Here are 5 related questions:
1. How do you apply the product rule in differentiation?
2. Can you differentiate \( \frac{5x^2 + 4x}{x^2 + 1} \) using the quotient rule?
3. How do you find the second derivative of \( \frac{3x + 2}{2x + 3} \)?
4. What is the chain rule in calculus, and when is it used?
5. How do you compute the derivative of a function raised to a power, like \( (2x+3)^5 \)?

**Tip:** Quotient rule is especially useful when dividing two functions, but if the denominator is constant, use the simpler constant multiple rule instead.

Learn how to differentiate the expression (3x + 2) / (2x + 3) using the quotient rule. This step-by-step solution covers key calculus concepts and formulas, including the quotient rule and the derivative of linear functions. Suitable for students in Grades 11-12.

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