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

We are required to locate the derivative.

Applying the rule of quotients, which is

where

This can be made simpler to

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To differentiate ( f(x) = \frac{e^x}{e^{x-2}-4} ) using the quotient rule, you first identify ( u ) as the numerator ( e^x ) and ( v ) as the denominator ( e^{x-2}-4 ). Then, apply the quotient rule formula:

[ f'(x) = \frac{u'v - uv'}{v^2} ]

Where:

- ( u' ) denotes the derivative of the numerator ( e^x ), which is ( e^x ).
- ( v' ) denotes the derivative of the denominator ( e^{x-2}-4 ).
- ( v^2 ) represents the square of the denominator.

Differentiate ( e^{x-2}-4 ) to get ( e^{x-2} ).

Then, apply the quotient rule formula:

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

Simplify the expression:

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

This is the derivative of the function ( f(x) ).

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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.

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.

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.

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