# How do you differentiate #y = arcsin (2x^3)#?

In this way, with the chain rule:

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To differentiate ( y = \arcsin(2x^3) ), you can use the chain rule. The derivative of ( \arcsin(u) ) with respect to ( u ) is ( \frac{1}{\sqrt{1 - u^2}} ), and then you multiply by the derivative of the inner function. So, applying the chain rule, the derivative of ( y ) with respect to ( x ) is:

[ \frac{d}{dx} \left( \arcsin(2x^3) \right) = \frac{1}{\sqrt{1 - (2x^3)^2}} \cdot \frac{d}{dx} (2x^3) ]

Simplify the expression:

[ \frac{d}{dx} \left( \arcsin(2x^3) \right) = \frac{1}{\sqrt{1 - 4x^6}} \cdot 6x^2 ]

Thus, the derivative of ( y = \arcsin(2x^3) ) with respect to ( x ) is:

[ \frac{6x^2}{\sqrt{1 - 4x^6}} ]

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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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