# How do you use the chain rule to differentiate #y=(x-1/x)^(3/2)#?

The chain rule states that:

So:

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To differentiate ( y = \left( x - \frac{1}{x} \right)^{\frac{3}{2}} ) using the chain rule, follow these steps:

- Let ( u = x - \frac{1}{x} ).
- Find ( \frac{du}{dx} ).
- Apply the power rule to differentiate ( u^{\frac{3}{2}} ) with respect to ( u ).
- Substitute ( \frac{du}{dx} ) and ( \frac{du}{dx} ) into the chain rule formula ( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} ).

The derivative will be:

[ \frac{dy}{dx} = \frac{3}{2}u^{\frac{1}{2}} \frac{du}{dx} ]

Finally, replace ( u ) with ( x - \frac{1}{x} ) and ( \frac{du}{dx} ) with its value.

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To differentiate ( y = \left(x - \frac{1}{x}\right)^{\frac{3}{2}} ) using the chain rule, first, identify the inner function ( u = x - \frac{1}{x} ).

Then, differentiate ( u ) with respect to ( x ):

( \frac{du}{dx} = 1 + \frac{1}{x^2} )

Next, differentiate ( y ) with respect to ( u ):

( \frac{dy}{du} = \frac{3}{2} \left(x - \frac{1}{x}\right)^{\frac{1}{2}} )

Finally, apply the chain rule:

( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} )

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