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

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To differentiate the given function ( y = -x^2(-3x^2 - 2) ) using the product rule, we follow these steps:

- Identify the two functions being multiplied: ( u = -x^2 ) and ( v = -3x^2 - 2 ).
- Apply the product rule formula ( (uv)' = u'v + uv' ).
- Find the derivatives of ( u ) and ( v ):
- ( u' = -2x ) (derivative of ( -x^2 )).
- ( v' = -6x ) (derivative of ( -3x^2 - 2 )).

- Substitute the derivatives and the original functions into the product rule formula:
- ( (uv)' = (-2x)(-3x^2 - 2) + (-x^2)(-6x) ).

- Simplify the expression:
- ( (-2x)(-3x^2 - 2) + (-x^2)(-6x) = 6x^3 + 4x + 6x^3 = 12x^3 + 4x ).

Therefore, the derivative of ( y = -x^2(-3x^2 - 2) ) with respect to ( x ) using the product rule is ( 12x^3 + 4x ).

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To differentiate y = -x²(-3x² - 2) using the product rule, first identify the two functions being multiplied: f(x) = -x² and g(x) = -3x² - 2. Then apply the product rule formula: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). Differentiate each function separately with respect to x, then apply the product rule formula to find the derivative of the product of the two functions.

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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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