Question

find the derivative the following ways. a. using the product rule. b. by expanding the product first. verify that your answer agrees with part (a). f(x)=(x - 6)(2x + 1) a. use the product rule to find the derivative of the function. select the correct choice and fill in the answer box(es) to complete your choice. a. the derivative is (x - 6) b. the derivative is (2x + 1)+(x - 6) c. the derivative is (x - 6)(2x + 1)+ d. the derivative is x(2x + 1) e. the derivative is (x - 6)(2x + 1)

Explanation:

Step1: Recall product - rule

The product rule states that if \(y = u\cdot v\), where \(u\) and \(v\) are functions of \(x\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u=x - 6\) and \(v = 2x+1\).

Step2: Find \(u^\prime\) and \(v^\prime\)

The derivative of \(u=x - 6\) with respect to \(x\) is \(u^\prime=1\), and the derivative of \(v = 2x+1\) with respect to \(x\) is \(v^\prime=2\).

Step3: Apply product - rule

\[

$$\begin{align*} f^\prime(x)&=u^\prime v+uv^\prime\\ &=1\cdot(2x + 1)+(x - 6)\cdot2\\ &=2x+1 + 2x-12\\ &=4x-11 \end{align*}$$

\]
Now, if we expand \(f(x)=(x - 6)(2x + 1)=2x^{2}+x-12x - 6=2x^{2}-11x - 6\).

Step4: Differentiate the expanded form

The derivative of \(y = 2x^{2}-11x - 6\) using the power - rule \((x^n)^\prime=nx^{n - 1}\) is \(y^\prime=4x-11\), which is the same as the result from the product - rule.

The derivative using the product rule: \(f^\prime(x)=(x - 6)\times2+(2x + 1)\times1\).

Answer:

C. The derivative is \((x - 6)(2)+(2x + 1)(1)\)