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How do you differentiate #y=(-2x^4+5x^2+4)(-3x^2+2)# using the product rule?

Answer 1

William Abdalla

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Answer 2

Charles Arocha

To differentiate ( y = (-2x^4 + 5x^2 + 4)(-3x^2 + 2) ) using the product rule:

Identify the two functions: ( f(x) = -2x^4 + 5x^2 + 4 ) and ( g(x) = -3x^2 + 2 ).
Apply the product rule: ( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) ).
Find the derivatives of ( f(x) ) and ( g(x) ):
- ( f'(x) = -8x^3 + 10x )
- ( g'(x) = -6x ).
Substitute into the product rule formula:
- ( y' = (-8x^3 + 10x)(-3x^2 + 2) + (-2x^4 + 5x^2 + 4)(-6x) ).
Simplify the expression to get the derivative.

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Answer from HIX Tutor

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.