How do you differentiate #f(x)= e^x/(e^(3-x) -8 )# using the quotient rule?
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To differentiate the function ( f(x) = \frac{e^x}{e^{3-x} - 8} ) using the quotient rule, follow these steps:
- Identify ( u(x) ) and ( v(x) ) as the numerator and denominator functions, respectively.
- Apply the quotient rule formula: [ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ]
- Differentiate ( u(x) ) and ( v(x) ) separately.
- Substitute the derivatives and original functions into the quotient rule formula.
- Simplify the expression if possible.
Let's denote ( u(x) = e^x ) and ( v(x) = e^{3-x} - 8 ).
The derivatives are: [ u'(x) = e^x ] [ v'(x) = -e^{3-x} ]
Now, substitute into the quotient rule formula: [ f'(x) = \frac{(e^x)(e^{3-x} - 8) - (e^x)(-e^{3-x})}{(e^{3-x} - 8)^2} ]
Finally, simplify the expression: [ f'(x) = \frac{e^x(e^{3-x} - 8 + e^{3-x})}{(e^{3-x} - 8)^2} ] [ f'(x) = \frac{e^x(2e^{3-x} - 8)}{(e^{3-x} - 8)^2} ]
Therefore, the derivative of ( f(x) ) with respect to ( x ) using the quotient rule is: [ f'(x) = \frac{e^x(2e^{3-x} - 8)}{(e^{3-x} - 8)^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.
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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