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)

c. the derivative is (x - 6)(2x + 1)+()

d. the derivative is ()x(2x + 1)

e. the derivative is (x - 6)(2x + 1)()

b. expand the product
(x - 6)(2x + 1)=2x^2 - 11x - 6 (simplify your answer)
select the correct choice and fill in the answer box(es) to complete your choice

a. the derivatives are different. using the product rule, the derivative is , and when expanding the product first, the derivative is

b. using either approach, \\(\frac{d}{dx}(x - 6)(2x + 1)=\\)

Explanation:

Step1: Recall product - rule

The product rule states that if $y = u(x)v(x)$, then $y^\prime=u^\prime(x)v(x)+u(x)v^\prime(x)$. Let $u(x)=x - 6$ and $v(x)=2x + 1$. Then $u^\prime(x)=1$ and $v^\prime(x)=2$.

Step2: Apply product - rule

Using the product rule, $f^\prime(x)=(x - 6)^\prime(2x + 1)+(x - 6)(2x + 1)^\prime=1\times(2x + 1)+(x - 6)\times2=2x+1 + 2x-12=4x - 11$.

Step3: Expand the product

Expand $f(x)=(x - 6)(2x + 1)=2x^{2}+x-12x - 6=2x^{2}-11x - 6$.

Step4: Differentiate the expanded form

Differentiate $y = 2x^{2}-11x - 6$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$. So $y^\prime=\frac{d}{dx}(2x^{2})-\frac{d}{dx}(11x)-\frac{d}{dx}(6)=4x-11$.

Answer:

B. Using either approach, $\frac{d}{dx}(x - 6)(2x + 1)=4x - 11$