# How do you differentiate #3sin^3(2x^2) # using the chain rule?

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To differentiate (3\sin^3(2x^2)) using the chain rule, follow these steps:

- Identify the outer function (f(u) = u^3) and the inner function (u = \sin(2x^2)).
- Find the derivative of the outer function with respect to (u), which is (f'(u) = 3u^2).
- Find the derivative of the inner function with respect to (x), which is (u' = \cos(2x^2) \cdot 4x).
- Substitute the expressions for (f'(u)) and (u') into the chain rule formula: [ \frac{d}{dx}[f(u)] = f'(u) \cdot u' ]
- Plug in the expressions for (f'(u)) and (u'), yielding: [ \frac{d}{dx}[3\sin^3(2x^2)] = 3(2\sin(2x^2))^2 \cdot \cos(2x^2) \cdot 4x ]
- Simplify the expression as needed.

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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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