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

Answer 1

Emma Aldridge

#f'(x) = 2frac{1-e^x}{(e^{-x}-2)^2} #

#f'(x) = frac{d}{dx}(frac{e^x}{e^{-x}-2})#

#= frac{(e^{-x}-2)frac{d}{dx}(e^x)-e^xfrac{d}{dx}(e^{-x}-2)}{(e^{-x}-2)^2} #

#= frac{(e^{-x}-2)e^x-e^x(-e^{-x})}{(e^{-x}-2)^2} #

#= frac{1-2e^x+1}{(e^{-x}-2)^2} #

#= 2frac{1-e^x}{(e^{-x}-2)^2} #

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Answer 2

Emily Ammerman

To differentiate the function f(x) = e^x / (e^(-x) - 2) using the quotient rule, you would follow these steps:

Identify the numerator and denominator functions.
Apply the quotient rule, which states that for a function u(x) / v(x), the derivative is (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2.
Find the derivatives of the numerator and denominator.
Plug these derivatives into the quotient rule formula.
Simplify the resulting expression if possible.

The derivative of f(x) using the quotient rule is given by:

f'(x) = (e^(-x) * e^x - e^x * (-e^(-x))) / (e^(-x) - 2)^2.

Simplify this expression to get the final derivative.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.