What is the derivative of #sqrt(x^2-1) / (x^2+1)#?

Answer 1

Christopher Allers

I found: #(x(3-x^2))/((x^2+1)^2sqrt(x^2-1))#

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Answer 2

Emma Aldridge

To find the derivative of the given function, we can use the quotient rule:

Given: ( f(x) = \frac{\sqrt{x^2 - 1}}{x^2 + 1} )

Let ( u = \sqrt{x^2 - 1} ) and ( v = x^2 + 1 ).

Now, ( u' = \frac{1}{2\sqrt{x^2 - 1}} \cdot (2x) ) by the chain rule, and ( v' = 2x ).

Applying the quotient rule:

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

[ f'(x) = \frac{\frac{1}{2\sqrt{x^2 - 1}} \cdot (2x) \cdot (x^2 + 1) - (\sqrt{x^2 - 1}) \cdot 2x}{(x^2 + 1)^2} ]

[ f'(x) = \frac{x(x^2 + 1) - 2x\sqrt{x^2 - 1}}{2\sqrt{x^2 - 1}(x^2 + 1)^2} ]

[ f'(x) = \frac{x^3 + x - 2x\sqrt{x^2 - 1}}{2\sqrt{x^2 - 1}(x^2 + 1)^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.