# How do you differentiate #f(x)= (x^3+2x+1)(2+x^-2)# using the product rule?

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To differentiate ( f(x) = (x^3 + 2x + 1)(2 + x^{-2}) ) using the product rule, you apply the formula:

[ \frac{d}{dx}(uv) = u'\cdot v + u \cdot v' ]

Where ( u = x^3 + 2x + 1 ) and ( v = 2 + x^{-2} ).

The derivatives are:

[ u' = 3x^2 + 2 ] [ v' = -2x^{-3} ]

Now, applying the product rule:

[ f'(x) = (3x^2 + 2)(2 + x^{-2}) + (x^3 + 2x + 1)(-2x^{-3}) ]

[ f'(x) = (6x^2 + 4) + \frac{3x^3}{x^2} - \frac{4x}{x^2} - \frac{2}{x^2} ]

[ f'(x) = 6x^2 + 4 + 3x - 4x^{-1} - 2x^{-2} ]

So, the derivative of ( f(x) ) is ( f'(x) = 6x^2 + 3x + 4 - \frac{4}{x} - \frac{2}{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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