find and simplify the derivative of the following function.\ng(x)=e^{x}(7x^{2}-14x + 14)\nwhich of the…

find and simplify the derivative of the following function.\ng(x)=e^{x}(7x^{2}-14x + 14)\nwhich of the following shows how to find the derivative of g(x)?\na. g(x)=(e^{x})(\\frac{d}{dx}(e^{x}))+(7x^{2}-14x + 14)(\\frac{d}{dx}(7x^{2}-14x + 14))\nb. g(x)=\\frac{(7x^{2}-14x + 14)(\\frac{d}{dx}(7x^{2}-14x + 14))+(e^{x})(\\frac{d}{dx}(e^{x}))}{(7x^{2}-14x + 14)^{2}}\nc. g(x)=(\\frac{d}{dx}(e^{x}))(7x^{2}-14x + 14)+(e^{x})(\\frac{d}{dx}(7x^{2}-14x + 14))\nd. g(x)=\\frac{(7x^{2}-14x + 14)(\\frac{d}{dx}(e^{x}))+(e^{x})(\\frac{d}{dx}(7x^{2}-14x + 14))}{(e^{x})^{2}}\nthe derivative of g(x)=e^{x}(7x^{2}-14x + 14) is \\square.

find and simplify the derivative of the following function.\ng(x)=e^{x}(7x^{2}-14x + 14)\nwhich of the following shows how to find the derivative of g(x)?\na. g(x)=(e^{x})(\\frac{d}{dx}(e^{x}))+(7x^{2}-14x + 14)(\\frac{d}{dx}(7x^{2}-14x + 14))\nb. g(x)=\\frac{(7x^{2}-14x + 14)(\\frac{d}{dx}(7x^{2}-14x + 14))+(e^{x})(\\frac{d}{dx}(e^{x}))}{(7x^{2}-14x + 14)^{2}}\nc. g(x)=(\\frac{d}{dx}(e^{x}))(7x^{2}-14x + 14)+(e^{x})(\\frac{d}{dx}(7x^{2}-14x + 14))\nd. g(x)=\\frac{(7x^{2}-14x + 14)(\\frac{d}{dx}(e^{x}))+(e^{x})(\\frac{d}{dx}(7x^{2}-14x + 14))}{(e^{x})^{2}}\nthe derivative of g(x)=e^{x}(7x^{2}-14x + 14) is \\square.

Answer

Explanation:

Step1: Apply the product rule

The product rule states that if (g(x)=u(x)v(x)), then (g^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)). Here, (u(x) = e^{x}) and (v(x)=7x^{2}-14x + 14). So, (g^{\prime}(x)=\left(\frac{d}{dx}(e^{x})\right)(7x^{2}-14x + 14)+(e^{x})\left(\frac{d}{dx}(7x^{2}-14x + 14)\right))

Step2: Calculate the derivatives of (u(x)) and (v(x))

  • The derivative of (u(x)=e^{x}) is (\frac{d}{dx}(e^{x})=e^{x})
  • The derivative of (v(x)=7x^{2}-14x + 14) is (\frac{d}{dx}(7x^{2}-14x + 14)=14x-14)

Step3: Substitute the derivatives into the product - rule formula

(g^{\prime}(x)=e^{x}(7x^{2}-14x + 14)+e^{x}(14x - 14)) Factor out (e^{x}): (g^{\prime}(x)=e^{x}[(7x^{2}-14x + 14)+(14x - 14)]) Simplify the expression inside the brackets: ((7x^{2}-14x + 14)+(14x - 14)=7x^{2})

Answer:

C. (g^{\prime}(x)=\left(\frac{d}{dx}(e^{x})\right)(7x^{2}-14x + 14)+(e^{x})\left(\frac{d}{dx}(7x^{2}-14x + 14)\right)) The derivative of (g(x)=e^{x}(7x^{2}-14x + 14)) is (7x^{2}e^{x})