# How do you differentiate #g(z) = z^3sin^2(2z)# using the product rule?

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To differentiate ( g(z) = z^3\sin^2(2z) ) using the product rule:

- Identify the two functions being multiplied: ( f(z) = z^3 ) and ( h(z) = \sin^2(2z) ).
- Apply the product rule: ( g'(z) = f'(z)h(z) + f(z)h'(z) ).
- Find the derivatives of ( f(z) ) and ( h(z) ):
- ( f'(z) = 3z^2 ) (using the power rule)
- ( h'(z) = 2\sin(2z)\cos(2z) ) (using the chain rule)

- Substitute these derivatives into the product rule formula: ( g'(z) = 3z^2 \sin^2(2z) + z^3(2\sin(2z)\cos(2z)) ).
- Simplify the expression if necessary, which may involve applying trigonometric identities.

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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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