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

Answer 1

Evelyn Agard

#g'(x)=4x^3-2x#

#g(x)=x^2(x^2-1)# Differentiating Both sides with respect to #x# we get #g'(x)=2x (x^2-1)+x^2×2x# #:.g'(x)=4x^3-2x#

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Answer 2

Christopher Agresta

To differentiate the function ( g(y) = x^2(x^2 - 1) ) using the product rule:

Identify the two functions being multiplied: ( f(x) = x^2 ) and ( h(x) = x^2 - 1 ).
Apply the product rule formula: ( g'(x) = f'(x)h(x) + f(x)h'(x) ).
Find the derivatives of ( f(x) ) and ( h(x) ): ( f'(x) = 2x ) and ( h'(x) = 2x ).
Substitute these derivatives and the original functions into the product rule formula.
Simplify the expression to get the derivative of ( g(y) ), which is ( g'(y) ).

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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.