# How do you find the derivative of # arcsin(x^2)#?

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

Differentiate Implicitly:

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To find the derivative of ( \arcsin(x^2) ), you can use the chain rule, which states that if ( f(u) ) is differentiable and ( g(x) ) is differentiable, then ( (f(g(x)))' = f'(g(x)) \cdot g'(x) ).

Let ( f(u) = \arcsin(u) ) and ( g(x) = x^2 ).

Then ( f'(u) = \frac{1}{\sqrt{1 - u^2}} ) and ( g'(x) = 2x ).

Using the chain rule, the derivative of ( \arcsin(x^2) ) is ( \frac{1}{\sqrt{1 - (x^2)^2}} \cdot 2x ), which simplifies to ( \frac{2x}{\sqrt{1 - x^4}} ).

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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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