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

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

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

Answer

Explanation:

Step1: Recall quotient - rule

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

Step2: Identify correct formula

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

Answer:

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