differentiate the following function.\ny = 8 ln x + ln 8\nchoose the correct setup below to start…

differentiate the following function.\ny = 8 ln x + ln 8\nchoose the correct setup below to start differentiating the function.\na. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { d } { d x } ( ln ( 8 x + 8 ) ) \\)\nb. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { d } { d x } ( 8 ln x ) + \\frac { d } { d x } ( ln 8 ) \\)\nc. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { ln 8 cdot \\frac { d } { d x } ( 8 ln x ) + 8 ln x cdot \\frac { d } { d x } ( ln 8 ) } { ( ln 8 ) ^ { 2 } } \\)\nd. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = ln 8 cdot \\frac { d } { d x } ( 8 ln x ) + 8 ln x cdot \\frac { d } { d x } ( ln 8 ) \\)\n\\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\)

differentiate the following function.\ny = 8 ln x + ln 8\nchoose the correct setup below to start differentiating the function.\na. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { d } { d x } ( ln ( 8 x + 8 ) ) \\)\nb. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { d } { d x } ( 8 ln x ) + \\frac { d } { d x } ( ln 8 ) \\)\nc. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\frac { ln 8 cdot \\frac { d } { d x } ( 8 ln x ) + 8 ln x cdot \\frac { d } { d x } ( ln 8 ) } { ( ln 8 ) ^ { 2 } } \\)\nd. \\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = ln 8 cdot \\frac { d } { d x } ( 8 ln x ) + 8 ln x cdot \\frac { d } { d x } ( ln 8 ) \\)\n\\( \\frac { d } { d x } ( 8 ln x + ln 8 ) = \\)

Answer

Explanation:

Step1: Apply sum rule of differentiation

The sum rule states that (\frac{d}{dx}(u + v)=\frac{d}{dx}(u)+\frac{d}{dx}(v)). For (y = 8\ln x+\ln8), let (u = 8\ln x) and (v=\ln8). So (\frac{d}{dx}(8\ln x+\ln8)=\frac{d}{dx}(8\ln x)+\frac{d}{dx}(\ln8))

Step2: Differentiate (8\ln x)

Using the constant - multiple rule (\frac{d}{dx}(k\cdot f(x))=k\cdot\frac{d}{dx}(f(x))) and the formula (\frac{d}{dx}(\ln x)=\frac{1}{x}), we have (\frac{d}{dx}(8\ln x)=8\cdot\frac{d}{dx}(\ln x)=8\cdot\frac{1}{x}=\frac{8}{x})

Step3: Differentiate (\ln8)

Since (\ln8) is a constant, and the derivative of a constant (C) is (0) ((\frac{d}{dx}(C) = 0)), so (\frac{d}{dx}(\ln8)=0)

Answer:

(\frac{8}{x})