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

Answer 1

Charles Anctil

#(-4e^x)/(e^x-4)^2#

Here it is #((e^x -4)d/dx e^x - e^x d/dx (e^x-4))/(e^x -4)^2#

=#(-4e^x)/(e^x-4)^2#

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Answer 2

Christopher Allers

To differentiate the function ( f(x) = \frac{e^x}{e^x - 4} ) using the quotient rule, you first identify the numerator and denominator functions: ( u(x) = e^x ) and ( v(x) = e^x - 4 ). Then, apply the quotient rule formula, which states that the derivative of a quotient of two functions is equal to (\frac{u'v - uv'}{v^2}). Substituting the functions and their derivatives into the formula, we get:

[ f'(x) = \frac{(e^x)(e^x - 4) - (e^x)(e^x)}{(e^x - 4)^2} ]

[ f'(x) = \frac{(e^x)(e^x - 4 - e^x)}{(e^x - 4)^2} ]

[ f'(x) = \frac{(e^x)(-4)}{(e^x - 4)^2} ]

[ f'(x) = \frac{-4e^x}{(e^x - 4)^2} ]

So, the derivative of ( f(x) ) with respect to ( x ) using the quotient rule is ( \frac{-4e^x}{(e^x - 4)^2} ).

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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.