# How do you differentiate #f(x)= e^x/(x-3 )# using the quotient rule?

As per the quotient rule,

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To differentiate ( f(x) = \frac{e^x}{x - 3} ) using the quotient rule, follow these steps:

- Identify ( u(x) ) as the numerator ( e^x ) and ( v(x) ) as the denominator ( x - 3 ).
- Compute ( u'(x) ) and ( v'(x) ), which are the derivatives of ( u(x) ) and ( v(x) ) respectively.
- Apply the quotient rule formula: [ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]
- Substitute the values of ( u(x) ), ( v(x) ), ( u'(x) ), and ( v'(x) ) into the formula.
- Simplify the expression if necessary.

Applying the quotient rule to the given function, we get:

[ f'(x) = \frac{(e^x)(x - 3) - (e^x)(1)}{(x - 3)^2} ]

[ = \frac{e^x(x - 3) - e^x}{(x - 3)^2} ]

[ = \frac{e^x(x - 3 - 1)}{(x - 3)^2} ]

[ = \frac{e^x(x - 4)}{(x - 3)^2} ]

So, the derivative of ( f(x) = \frac{e^x}{x - 3} ) using the quotient rule is ( f'(x) = \frac{e^x(x - 4)}{(x - 3)^2} ).

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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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