# How do you differentiate #f(x) =1/(e^(3-x)+1)# using the quotient rule?

According to the quotient rule,

Finding both of the internal derivatives, we see that

To find the second derivative, use the chain rule:

Plug these both back in:

We can try to simplify further:

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To differentiate ( f(x) = \frac{1}{e^{3-x} + 1} ) using the quotient rule, follow these steps:

- Identify ( u(x) ) as the numerator and ( v(x) ) as the denominator.
- Compute ( u'(x) ) and ( v'(x) ).
- Apply the quotient rule formula: [ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} ]
- Substitute the values obtained into the formula.

So, the differentiation of ( f(x) ) with respect to ( x ) using the quotient rule is: [ f'(x) = \frac{(0)(e^{3-x} + 1) - (1)(-e^{3-x})}{(e^{3-x} + 1)^2} ] [ f'(x) = \frac{e^{3-x}}{(e^{3-x} + 1)^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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