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

Answer 1

Scarlett Allison

See the answer below:

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Answer 2

Charles Arocha

To differentiate ( f(x) = \frac{e^{x/2}}{2x^3} ) using the quotient rule:

Identify the numerator function ( u(x) = e^{x/2} ) and the denominator function ( v(x) = 2x^3 ).
Apply the quotient rule: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).
Calculate the derivatives:
- ( u'(x) = \frac{1}{2} e^{x/2} ) (using the chain rule)
- ( v'(x) = 6x^2 ) (using the power rule)
Substitute the derivatives and functions into the quotient rule formula and simplify.

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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.