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We are given the function \( k(x) = \frac{f(x)g(x)}{h(x)} \), with the functions:
- \( f(x) = 3x + 2 \),
- \( g(x) = 3x - 4 \),
- \( h(x) = -x^2 - 2x + 4 \).

The task is to find \( k'(1) \).

To solve this, we will apply the quotient rule for derivatives. The quotient rule states:

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

Here:
- \( u(x) = f(x)g(x) \),
- \( v(x) = h(x) \).

### Step 1: Differentiate \( u(x) = f(x)g(x) \)
To find \( u'(x) \), we use the product rule:

\[
u'(x) = f'(x)g(x) + f(x)g'(x)
\]

First, we calculate the individual derivatives:
- \( f'(x) = 3 \) (since \( f(x) = 3x + 2 \)),
- \( g'(x) = 3 \) (since \( g(x) = 3x - 4 \)).

Thus,
\[
u'(x) = 3(3x - 4) + (3x + 2)(3)
= 9x - 12 + 9x + 6
= 18x - 6.
\]

### Step 2: Differentiate \( h(x) = -x^2 - 2x + 4 \)
\[
h'(x) = -2x - 2.
\]

### Step 3: Apply the quotient rule
Using the quotient rule formula:

\[
k'(x) = \frac{(f'(x)g(x) + f(x)g'(x))h(x) - f(x)g(x)h'(x)}{h(x)^2}
= \frac{u'(x)h(x) - u(x)h'(x)}{h(x)^2}.
\]

At \( x = 1 \), we need to evaluate each function:

- \( f(1) = 3(1) + 2 = 5 \),
- \( g(1) = 3(1) - 4 = -1 \),
- \( h(1) = -(1)^2 - 2(1) + 4 = 1 \),
- \( u'(1) = 18(1) - 6 = 12 \),
- \( h'(1) = -2(1) - 2 = -4 \).

Now calculate \( u(1) \) (which is \( f(1)g(1) \)):
\[
u(1) = 5(-1) = -5.
\]

Finally, substitute these into the quotient rule expression:

\[
k'(1) = \frac{12(1) - (-5)(-4)}{1^2} = \frac{12 - 20}{1} = -8.
\]

Thus, \( k'(1) = -8 \).

The correct answer is \( \boxed{-8} \).

Would you like more details on any step?

Here are five related questions you might consider:
1. How would the answer change if \( h(x) \) were a linear function?
2. What is the significance of the quotient rule in real-world applications?
3. How does the product rule relate to the chain rule in multivariable calculus?
4. How would you approach finding the second derivative \( k''(x) \)?
5. What are some common mistakes when applying the quotient rule?

**Tip:** Always verify the critical points of the denominator when applying the quotient rule to avoid undefined values in the final expression.

Let k(x) = (f(x)g(x))/h(x). If f(x) = 3x + 2, g(x) = 3x − 4, and h(x) = −x² − 2x + 4, what is k'(1)?

Learn how to find the derivative of a quotient function using the quotient rule. This calculus problem involves applying both the quotient and product rules, with detailed calculations of k'(1) for functions f(x), g(x), and h(x).

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