find the derivative of the function.\ny = 2x^6e^x\nwhich of the following shows how to find the derivative…

find the derivative of the function.\ny = 2x^6e^x\nwhich of the following shows how to find the derivative of the function?\na. y = 2x^6(\\frac{d}{dx}(2x^6))+(\\frac{d}{dx}(e^x))e^x\nb. y = 2x^6(\\frac{d}{dx}(e^x))+(\\frac{d}{dx}(2x^6))e^x\nc. y = \\frac{2x^6(\\frac{d}{dx}(e^x)) - e^x(\\frac{d}{dx}(2x^6))}{(2x^6)^2}\nd. y = \\frac{e^x(\\frac{d}{dx}(2x^6)) - 2x^6(\\frac{d}{dx}(e^x))}{(e^x)^2}

find the derivative of the function.\ny = 2x^6e^x\nwhich of the following shows how to find the derivative of the function?\na. y = 2x^6(\\frac{d}{dx}(2x^6))+(\\frac{d}{dx}(e^x))e^x\nb. y = 2x^6(\\frac{d}{dx}(e^x))+(\\frac{d}{dx}(2x^6))e^x\nc. y = \\frac{2x^6(\\frac{d}{dx}(e^x)) - e^x(\\frac{d}{dx}(2x^6))}{(2x^6)^2}\nd. y = \\frac{e^x(\\frac{d}{dx}(2x^6)) - 2x^6(\\frac{d}{dx}(e^x))}{(e^x)^2}

Answer

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'=u\cdot v'+v\cdot u'). Here, (u = 2x^{6}) and (v = e^{x}).

Step2: Identify (u), (v), (u') and (v')

We know that (\frac{d}{dx}(2x^{6})=12x^{5}) (using the power - rule (\frac{d}{dx}(ax^{n})=nax^{n - 1})) and (\frac{d}{dx}(e^{x})=e^{x}). By the product - rule (y'=(2x^{6})\cdot\frac{d}{dx}(e^{x})+e^{x}\cdot\frac{d}{dx}(2x^{6})).

Answer:

B. (y' = 2x^{6}\left(\frac{d}{dx}(e^{x})\right)+\left(\frac{d}{dx}(2x^{6})\right)e^{x})