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

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To differentiate ( f(x) = (x^2 + 1)(x^3 + 1) ) using the product rule, you would follow these steps:

- Identify the two functions being multiplied: ( u = x^2 + 1 ) and ( v = x^3 + 1 ).
- Apply the product rule: ( f'(x) = u'v + uv' ).
- Find the derivatives of ( u ) and ( v ): ( u' = 2x ) and ( v' = 3x^2 ).
- Substitute the derivatives and the original functions into the product rule formula: ( f'(x) = (2x)(x^3 + 1) + (x^2 + 1)(3x^2) ).
- Simplify the expression: ( f'(x) = 2x^4 + 2x + 3x^4 + 3x^2 ).
- Combine like terms: ( f'(x) = 5x^4 + 3x^2 + 2x ).

Therefore, the derivative of ( f(x) = (x^2 + 1)(x^3 + 1) ) using the product rule is ( f'(x) = 5x^4 + 3x^2 + 2x ).

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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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