# How do you differentiate #g(x) = sqrt(x^3-4)cos2x# using the product rule?

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To differentiate ( g(x) = \sqrt{x^3 - 4} \cos(2x) ) using the product rule, follow these steps:

- Identify the two functions within the product: ( f(x) = \sqrt{x^3 - 4} ) and ( h(x) = \cos(2x) ).
- Apply the product rule: ( g'(x) = f(x)h'(x) + f'(x)h(x) ).
- Find the derivatives of ( f(x) ) and ( h(x) ):
- ( f'(x) = \frac{1}{2\sqrt{x^3 - 4}} \cdot \frac{d}{dx}(x^3 - 4) ).
- ( h'(x) = -2\sin(2x) ).

- Plug the derivatives and original functions into the product rule formula and simplify to get ( g'(x) ).

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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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