find the derivative the following ways. a. using the product rule. b. by expanding the product first. verify…

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))