Find the derivative of f(x)=
4x^2/
x^2+3x+2
.

To find the derivative of the function \( f(x) = \frac{4x^2}{x^2 + 3x + 2} \), we can apply the quotient rule. The quotient rule is given by:

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

For the given function, we have:
- \( u(x) = 4x^2 \)
- \( v(x) = x^2 + 3x + 2 \)

Let's find the derivatives \( u'(x) \) and \( v'(x) \):

\[
u'(x) = \frac{d}{dx}(4x^2) = 8x
\]
\[
v'(x) = \frac{d}{dx}(x^2 + 3x + 2) = 2x + 3
\]

Now, applying the quotient rule:

\[
f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}
\]
\[
f'(x) = \frac{(8x)(x^2 + 3x + 2) - (4x^2)(2x + 3)}{(x^2 + 3x + 2)^2}
\]

Now, let's expand and simplify the numerator:

\[
(8x)(x^2 + 3x + 2) = 8x^3 + 24x^2 + 16x
\]
\[
(4x^2)(2x + 3) = 8x^3 + 12x^2
\]
\[
\text{Numerator: } 8x^3 + 24x^2 + 16x - (8x^3 + 12x^2) = 12x^2 + 16x
\]

So the derivative is:

\[
f'(x) = \frac{12x^2 + 16x}{(x^2 + 3x + 2)^2}
\]

This is the derivative of the function \( f(x) \).

Would you like more details or have any questions?

Here are some related questions:
1. How would you find the second derivative of \( f(x) \)?
2. Can the quotient rule be applied to non-rational functions?
3. What are the alternative methods to differentiate this function?
4. How would you find the critical points of \( f(x) \)?
5. How does the function behave as \( x \) approaches infinity?

**Tip:** Always simplify the expression as much as possible after applying the quotient rule to make further calculations easier.

Learn how to find the derivative of the function f(x) = 4x^2 / (x^2 + 3x + 2) using the quotient rule. Detailed step-by-step solution provided, including differentiation of numerator and denominator. Suitable for advanced high school students and college-level calculus learners.

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