Derivative of Quotient Function Using Quotient Rule

Math Problem Statement

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)?

We are given the function $k(x) = \frac{f(x)g(x)}{h(x)}$ , with the functions:

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:

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:

Thus,

= 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.

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Calculus
Derivative Rules
Quotient Rule
Product Rule

Quotient rule: (u(x)/v(x))' = (u'(x)v(x) - u(x)v'(x)) / v(x)²
Product rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)

Quotient Rule for Derivatives
Product Rule for Derivatives

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