# How do you differentiate #y=((x^2+1)/(x^2-1))^3#?

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To differentiate the function ( y = \left( \frac{x^2+1}{x^2-1} \right)^3 ), you can use the chain rule. First, find the derivative of the inner function ( \frac{x^2+1}{x^2-1} ), and then multiply it by the derivative of ( x^3 ).

The derivative of ( \frac{x^2+1}{x^2-1} ) can be found using the quotient rule, which states that for functions ( u(x) ) and ( v(x) ), ( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} ).

Let ( u(x) = x^2+1 ) and ( v(x) = x^2-1 ). Then ( u'(x) = 2x ) and ( v'(x) = 2x ).

Using the quotient rule, ( \left( \frac{u}{v} \right)' = \frac{(2x)(x^2-1) - (x^2+1)(2x)}{(x^2-1)^2} ).

Simplify the expression.

After finding ( \frac{du}{dv} ), multiply it by the derivative of ( x^3 ), which is ( 3x^2 ).

So, ( \frac{dy}{dx} = 3x^2 \left( \frac{(2x)(x^2-1) - (x^2+1)(2x)}{(x^2-1)^2} \right) ).

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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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