find and simplify the derivative of the following function.\ng(x)=e^x(13x^2 - 26x + 26)\nwhich of the…

find and simplify the derivative of the following function.\ng(x)=e^x(13x^2 - 26x + 26)\nwhich of the following shows how to find the derivative of g(x)?\na. g(x)=(e^x)(\\frac{d}{dx}(e^x))+(13x^2 - 26x + 26)(\\frac{d}{dx}(13x^2 - 26x + 26))\nb. g(x)=\\frac{(13x^2 - 26x + 26)(\\frac{d}{dx}(13x^2 - 26x + 26))+(e^x)(\\frac{d}{dx}(e^x))}{(13x^2 - 26x + 26)^2}\nc. g(x)=(\\frac{d}{dx}(e^x))(13x^2 - 26x + 26)+(e^x)(\\frac{d}{dx}(13x^2 - 26x + 26))\nd. g(x)=\\frac{(13x^2 - 26x + 26)(\\frac{d}{dx}(e^x))+(e^x)(\\frac{d}{dx}(13x^2 - 26x + 26))}{(e^x)^2}\nthe derivative of g(x)=e^x(13x^2 - 26x + 26) is

find and simplify the derivative of the following function.\ng(x)=e^x(13x^2 - 26x + 26)\nwhich of the following shows how to find the derivative of g(x)?\na. g(x)=(e^x)(\\frac{d}{dx}(e^x))+(13x^2 - 26x + 26)(\\frac{d}{dx}(13x^2 - 26x + 26))\nb. g(x)=\\frac{(13x^2 - 26x + 26)(\\frac{d}{dx}(13x^2 - 26x + 26))+(e^x)(\\frac{d}{dx}(e^x))}{(13x^2 - 26x + 26)^2}\nc. g(x)=(\\frac{d}{dx}(e^x))(13x^2 - 26x + 26)+(e^x)(\\frac{d}{dx}(13x^2 - 26x + 26))\nd. g(x)=\\frac{(13x^2 - 26x + 26)(\\frac{d}{dx}(e^x))+(e^x)(\\frac{d}{dx}(13x^2 - 26x + 26))}{(e^x)^2}\nthe derivative of g(x)=e^x(13x^2 - 26x + 26) is

Answer

Explanation:

Step1: Recall product - rule

The product - rule states that if $y = u(x)v(x)$, then $y'=u'(x)v(x)+u(x)v'(x)$. Here, $u(x)=e^{x}$ and $v(x)=13x^{2}-26x + 26$.

Step2: Find derivatives of $u(x)$ and $v(x)$

The derivative of $u(x)=e^{x}$ with respect to $x$ is $\frac{d}{dx}(e^{x})=e^{x}$. The derivative of $v(x)=13x^{2}-26x + 26$ with respect to $x$ is $\frac{d}{dx}(13x^{2}-26x + 26)=26x-26$.

Step3: Apply product - rule

$g'(x)=\left(\frac{d}{dx}(e^{x})\right)(13x^{2}-26x + 26)+(e^{x})\left(\frac{d}{dx}(13x^{2}-26x + 26)\right)=e^{x}(13x^{2}-26x + 26)+e^{x}(26x - 26)$.

Step4: Simplify

Factor out $e^{x}$: $g'(x)=e^{x}(13x^{2}-26x + 26+26x - 26)=13x^{2}e^{x}$.

Answer:

C. $g^{\prime}(x)=\left(\frac{d}{dx}(e^{x})\right)(13x^{2}-26x + 26)+(e^{x})\left(\frac{d}{dx}(13x^{2}-26x + 26)\right)$; $13x^{2}e^{x}$