# How do you differentiate # y = sin 5x - 1/3 sin^3 5x#?

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To differentiate ( y = \sin(5x) - \frac{1}{3} \sin^3(5x) ), we use the chain rule and the power rule for differentiation.

The derivative of ( \sin(5x) ) is ( 5 \cos(5x) ) by the chain rule, and the derivative of ( \sin^3(5x) ) requires the chain rule and the power rule: ( 3 \sin^2(5x) \cos(5x) ).

Therefore, the derivative of ( y ) is:

[ y' = 5 \cos(5x) - \frac{1}{3} \times 3 \sin^2(5x) \cos(5x) ]

Simplifying further:

[ y' = 5 \cos(5x) - \sin^2(5x) \cos(5x) ]

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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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