# How do you differentiate #g(y) =(4x^2+5)(x^2 - 1) # using the product rule?

The product rule states that for a function

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To differentiate ( g(y) = (4x^2 + 5)(x^2 - 1) ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( f(x) = 4x^2 + 5 ) and ( h(x) = x^2 - 1 ).
- Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
- Differentiate each function separately: ( f'(x) = 8x ) (derivative of ( 4x^2 + 5 )) ( h'(x) = 2x ) (derivative of ( x^2 - 1 )).
- Apply the product rule formula: ( g'(x) = f'(x)h(x) + f(x)h'(x) ).
- Substitute the derivatives and original functions into the formula: ( g'(x) = (8x)(x^2 - 1) + (4x^2 + 5)(2x) ).
- Simplify the expression: ( g'(x) = 8x^3 - 8x + 8x^3 + 10x ).
- Combine like terms: ( g'(x) = 16x^3 + 2x ).

Therefore, the derivative of ( g(y) = (4x^2 + 5)(x^2 - 1) ) with respect to ( x ) using the product rule is ( g'(x) = 16x^3 + 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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