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How do you differentiate #f(x) =x^2/(xe^(1-x)+2)# using the quotient rule?

Answer 1

Charles Arocha

See the answer below:

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Answer 2

Cameron Alexander

To differentiate the function ( f(x) = \frac{x^2}{xe^{1-x} + 2} ) 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: ( 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.

Let's apply these steps:

( u(x) = x^2 ) and ( v(x) = xe^{1-x} + 2 ).
( u'(x) = 2x ) and ( v'(x) = e^{1-x} - xe^{1-x} ).
Using the quotient rule: ( f'(x) = \frac{(2x)(xe^{1-x} + 2) - (x^2)(e^{1-x} - xe^{1-x})}{(xe^{1-x} + 2)^2} ).
Substitute the values: ( f'(x) = \frac{2x(xe^{1-x} + 2) - x^2(e^{1-x} - xe^{1-x})}{(xe^{1-x} + 2)^2} ).
Simplify the expression if necessary.

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Answer from HIX Tutor

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.