# What is the derivative of this function #y=(cos^-1(4x^2))^2#?

Substituting,

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To find the derivative of the function ( y = (\cos^{-1}(4x^2))^2 ), we can use the chain rule and the power rule for differentiation.

Let ( u = \cos^{-1}(4x^2) ). Then ( y = u^2 ).

Using the chain rule, the derivative of ( u ) with respect to ( x ) is: [ \frac{du}{dx} = \frac{d}{dx}(\cos^{-1}(4x^2)) = -\frac{1}{\sqrt{1-(4x^2)^2}} \cdot \frac{d}{dx}(4x^2) ] [ = -\frac{1}{\sqrt{1-16x^4}} \cdot 8x ]

Now, using the power rule, the derivative of ( y ) with respect to ( u ) is: [ \frac{dy}{du} = \frac{d}{du}(u^2) = 2u ]

Finally, we combine these derivatives using the chain rule: [ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \left(-\frac{1}{\sqrt{1-16x^4}} \cdot 8x\right) ] [ = -\frac{16xu}{\sqrt{1-16x^4}} ]

Substituting back ( u = \cos^{-1}(4x^2) ): [ = -\frac{16x\cos^{-1}(4x^2)}{\sqrt{1-16x^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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