# How do you differentiate #f(x) =x^2/(e^(3-x)+2)# using the quotient rule?

then rearranging the numerator

God bless....I hope the explanation is useful.

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To differentiate the function ( f(x) = \frac{x^2}{e^{3-x} + 2} ) using the quotient rule, follow these steps:

- Identify ( u(x) = x^2 ) and ( v(x) = e^{3-x} + 2 ).
- Calculate ( u'(x) ) and ( v'(x) ).
- 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 to get the derivative of ( f(x) ).

The result of differentiating ( f(x) ) using the quotient rule will give you the derivative of the function.

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To differentiate ( f(x) = \frac{x^2}{e^{3-x} + 2} ) using the quotient rule, we apply the formula:

[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} ]

Where ( u = x^2 ) and ( v = e^{3-x} + 2 ). The derivatives are:

[ u' = 2x ] [ v' = -e^{3-x} ]

Now, we substitute these derivatives into the quotient rule formula:

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

Simplify this expression:

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

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

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