# What is the derivative of #y= ln(1-x^2)^(1/2)#?

Differentiate the left hand side of the equation using implicit differentiation. Differentiate the right hand side of the equation using the rule:

*Continued...

So what you end up with is...

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To find the derivative of ( y = \ln\left((1 - x^2)^{1/2}\right) ), we use the chain rule:

[ \frac{dy}{dx} = \frac{d}{dx}\left[\ln\left((1 - x^2)^{1/2}\right)\right] ]

Using the chain rule and the derivative of natural logarithm:

[ \frac{dy}{dx} = \frac{1}{(1 - x^2)^{1/2}} \cdot \frac{d}{dx}\left((1 - x^2)^{1/2}\right) ]

Now, we apply the power rule and chain rule to ( (1 - x^2)^{1/2} ):

[ \frac{dy}{dx} = \frac{1}{(1 - x^2)^{1/2}} \cdot \frac{1}{2}(1 - x^2)^{-1/2} \cdot \frac{d}{dx}(1 - x^2) ]

[ \frac{dy}{dx} = \frac{1}{(1 - x^2)} \cdot (-x) ]

[ \frac{dy}{dx} = \frac{-x}{(1 - x^2)} ]

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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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