find the derivative the following ways\na. using the product rule\nb. by expanding the product first. verify…

find the derivative the following ways\na. using the product rule\nb. by expanding the product first. verify that your answer agrees with part (a)\nf(x)=(x - 6)(2x + 1)\n\nc. the derivative is (x - 6)(2x + 1)+()\n\nd. the derivative is ()x(2x + 1)\n\ne. the derivative is (x - 6)(2x + 1)()\n\nb. expand the product\n(x - 6)(2x + 1)=2x^2 - 11x - 6 (simplify your answer)\nselect the correct choice and fill in the answer box(es) to complete your choice\n\na. the derivatives are different. using the product rule, the derivative is , and when expanding the product first, the derivative is \n\nb. using either approach, \\(\\frac{d}{dx}(x - 6)(2x + 1)=\\)

find the derivative the following ways\na. using the product rule\nb. by expanding the product first. verify that your answer agrees with part (a)\nf(x)=(x - 6)(2x + 1)\n\nc. the derivative is (x - 6)(2x + 1)+()\n\nd. the derivative is ()x(2x + 1)\n\ne. the derivative is (x - 6)(2x + 1)()\n\nb. expand the product\n(x - 6)(2x + 1)=2x^2 - 11x - 6 (simplify your answer)\nselect the correct choice and fill in the answer box(es) to complete your choice\n\na. the derivatives are different. using the product rule, the derivative is , and when expanding the product first, the derivative is \n\nb. using either approach, \\(\\frac{d}{dx}(x - 6)(2x + 1)=\\)

Answer

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, $\frac{d}{dx}(x - 6)(2x + 1)=4x - 11$