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

Answer 1

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

Using the Quotient rule we get #f'(x)=(e^x(e^(-x)-x)-e^x(-e^(-x)-1))/(e^(-x)-x)^2# Expanding and collect equal Terms we get the result above.

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Answer 2

Elijah Abram

To differentiate ( f(x) = \frac{e^x}{e^{-x} - x} ) using the quotient rule:

Apply the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} ).
Let ( u = e^x ) and ( v = e^{-x} - x ).
Find the derivatives: ( u' = e^x ) and ( v' = -e^{-x} - 1 ).
Substitute the derivatives and functions into the quotient rule formula.
Simplify the expression if possible.

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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.