find and simplify the derivative of the following function. f(x) = \\frac{3e^{x}}{e^{x}-8}\nwhich of the…

find and simplify the derivative of the following function. f(x) = \\frac{3e^{x}}{e^{x}-8}\nwhich of the following shows how to find the derivative of f(x)?\na. f(x) = \\frac{(e^{x}-8)(\\frac{d}{dx}(3e^{x}))-(3e^{x})(\\frac{d}{dx}(e^{x}-8))}{(e^{x}-8)^{2}}\nb. f(x)=(e^{x}-8)(\\frac{d}{dx}(3e^{x}))-(3e^{x})(\\frac{d}{dx}(e^{x}-8))\nc. f(x)=(3e^{x})(\\frac{d}{dx}(3e^{x}))+(e^{x}-8)(\\frac{d}{dx}(e^{x}-8))\nd. f(x)=\\frac{(3e^{x})(\\frac{d}{dx}(3e^{x}))+(e^{x}-8)(\\frac{d}{dx}(e^{x}-8))}{(e^{x}-8)}

find and simplify the derivative of the following function. f(x) = \\frac{3e^{x}}{e^{x}-8}\nwhich of the following shows how to find the derivative of f(x)?\na. f(x) = \\frac{(e^{x}-8)(\\frac{d}{dx}(3e^{x}))-(3e^{x})(\\frac{d}{dx}(e^{x}-8))}{(e^{x}-8)^{2}}\nb. f(x)=(e^{x}-8)(\\frac{d}{dx}(3e^{x}))-(3e^{x})(\\frac{d}{dx}(e^{x}-8))\nc. f(x)=(3e^{x})(\\frac{d}{dx}(3e^{x}))+(e^{x}-8)(\\frac{d}{dx}(e^{x}-8))\nd. f(x)=\\frac{(3e^{x})(\\frac{d}{dx}(3e^{x}))+(e^{x}-8)(\\frac{d}{dx}(e^{x}-8))}{(e^{x}-8)}

Answer

Explanation:

Step1: Recall quotient - rule

The quotient - rule for a function $y=\frac{u}{v}$ is $y'=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$, where $u = 3e^{x}$ and $v=e^{x}-8$.

Step2: Identify the correct formula

Applying the quotient - rule to $f(x)=\frac{3e^{x}}{e^{x}-8}$, we get $f'(x)=\frac{(e^{x}-8)\frac{d}{dx}(3e^{x})-(3e^{x})\frac{d}{dx}(e^{x}-8)}{(e^{x}-8)^{2}}$.

Answer:

A. $\frac{(e^{x}-8)\left(\frac{d}{dx}(3e^{x})\right)-(3e^{x})\left(\frac{d}{dx}(e^{x}-8)\right)}{(e^{x}-8)^{2}}$