# How do you differentiate #H(x)=(x+x^-1)^3#?

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To differentiate ( H(x) = (x + x^{-1})^3 ), you would use the chain rule and the power rule.

First, apply the chain rule:

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

Then, apply the power rule:

[ \frac{d}{dx} (x + x^{-1})^3 = 3(x + x^{-1})^2 \cdot \frac{d}{dx} (x + x^{-1}) ]

Substitute the derivative of ( (x + x^{-1}) ) from the chain rule:

[ \frac{d}{dx} (x + x^{-1})^3 = 3(x + x^{-1})^2 \cdot (1 - x^{-2}) ]

This simplifies to:

[ \frac{d}{dx} (x + x^{-1})^3 = 3(x^2 + 1 + x^{-2}) ]

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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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