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How do you find dy/dx using the quotient rule for #[e^(1/2x)]/(2x^3)#?

Answer 1

Scarlett Allison

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Answer 2

Charles Arocha

To find ( \frac{dy}{dx} ) using the quotient rule for ( \frac{e^{\frac{1}{2}x}}{2x^3} ), differentiate the numerator and denominator separately, then apply the quotient rule:

Differentiate the numerator: ( \frac{d}{dx}(e^{\frac{1}{2}x}) = \frac{1}{2}e^{\frac{1}{2}x} )
Differentiate the denominator: ( \frac{d}{dx}(2x^3) = 6x^2 )
Apply the quotient rule: ( \frac{d}{dx}\left(\frac{e^{\frac{1}{2}x}}{2x^3}\right) = \frac{(2x^3 \cdot \frac{1}{2}e^{\frac{1}{2}x}) - (e^{\frac{1}{2}x} \cdot 6x^2)}{(2x^3)^2} )
Simplify the expression.

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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.