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How do you differentiate #f(x)= ( 4- x )/ (e^x + 2) # using the quotient rule?

Answer 1

Scarlett Allison

#(df(x))/(dx)=(e^x(x-5)-2)/(e^x+2)^2#.

According to the quotient rule,

#(dg(x)/(h(x)))/(dx)=((dg(x))/(dx)*h(x)-g(x)(dh(x))/(dx))/(h(x))^2#.

In this case #g(x)=4-x#, #(dg(x))/(dx)=-1#, #h(x)=e^x+2#, and #(dh(x))/(dx)=e^x#, so

#(df(x))/(dx)=((-1)(e^x+2)-(4-x)e^x)/(e^x+2)^2=(e^x(x-5)-2)/(e^x+2)^2#.

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Answer 2

Ezekiel Alexander

To differentiate ( f(x) = \frac{4 - x}{e^x + 2} ) using the quotient rule, you apply the formula ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2} ), where ( u = 4 - x ) and ( v = e^x + 2 ). Then, ( u' = -1 ) and ( v' = e^x ). Substituting into the formula:

( f'(x) = \frac{(e^x + 2)(-1) - (4 - x)(e^x)}{(e^x + 2)^2} )

( f'(x) = \frac{-e^x - 2 - 4e^x + xe^x}{(e^x + 2)^2} )

( f'(x) = \frac{-5e^x - 2 + xe^x}{(e^x + 2)^2} )

So, the derivative of ( f(x) ) with respect to ( x ) is ( \frac{-5e^x - 2 + xe^x}{(e^x + 2)^2} ).

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Answer from HIX Tutor

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