# How do you find the derivative of #(x)/sqrt(x^2-4)#?

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To find the derivative of ( \frac{x}{\sqrt{x^2 - 4}} ), you can use the quotient rule, which states that if you have a function of the form ( \frac{u}{v} ), then the derivative is given by:

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

Let ( u = x ) and ( v = \sqrt{x^2 - 4} ). Then:

[ u' = 1 ] [ v' = \frac{1}{2}(x^2 - 4)^{-\frac{1}{2}}(2x) = \frac{x}{\sqrt{x^2 - 4}} ]

Now, apply the quotient rule:

[ \left(\frac{x}{\sqrt{x^2 - 4}}\right)' = \frac{(1)(\sqrt{x^2 - 4}) - (x)(\frac{x}{\sqrt{x^2 - 4}})}{(\sqrt{x^2 - 4})^2} ] [ = \frac{\sqrt{x^2 - 4} - \frac{x^2}{\sqrt{x^2 - 4}}}{x^2 - 4} ] [ = \frac{\sqrt{x^2 - 4} - \frac{x^2}{\sqrt{x^2 - 4}}}{x^2 - 4} ]

So, the derivative of ( \frac{x}{\sqrt{x^2 - 4}} ) is ( \frac{\sqrt{x^2 - 4} - \frac{x^2}{\sqrt{x^2 - 4}}}{x^2 - 4} ).

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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.

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.

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.

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