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What is the derivative of #x/(x^2-9)#?

Answer 1

Emilia Aguilar

Error on my part

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Answer 2

Paisley Johnson

To find the derivative of ( \frac{x}{x^2 - 9} ), you can use the quotient rule.

Let ( u = x ) and ( v = x^2 - 9 ).

Then, using the quotient rule, the derivative ( \frac{d}{dx} \left( \frac{u}{v} \right) ) is given by:

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

Plugging in ( u = x ) and ( v = x^2 - 9 ), and finding the derivatives:

[ \frac{du}{dx} = 1 ] [ \frac{dv}{dx} = 2x ]

Substituting into the quotient rule formula:

[ \frac{d}{dx} \left( \frac{x}{x^2 - 9} \right) = \frac{(x^2 - 9)(1) - x(2x)}{(x^2 - 9)^2} ]

Simplify the numerator:

[ (x^2 - 9) - 2x^2 = -x^2 - 9 ]

So, the derivative is:

[ \frac{-x^2 - 9}{(x^2 - 9)^2} ]

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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.