# How do I find the derivative of # y=s*sqrt(1-s^2) + cos^(-1)(s)#?

It is

#(d(y(s)))/(ds)=sqrt(1-s^2)+s*((-2s)/(2*sqrt(1-s^2)))-1/(sqrt(1-s^2))= sqrt(1-s^2)-((1+s^2)/(sqrt(1-s^2)))=(((sqrt(1-s^2))^2)-(1+s^2))/(sqrt(1-s^2))= ((1-s^2)-(1+s^2))/(sqrt(1-s^2))=-(2s^2)/(sqrt(1-s^2))#

Remarks

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To find the derivative of ( y = s \sqrt{1 - s^2} + \cos^{-1}(s) ), you can use the chain rule and the derivative of inverse trigonometric functions. The derivative is:

[ \frac{dy}{ds} = \sqrt{1 - s^2} - \frac{s^2}{\sqrt{1 - s^2}} - \frac{1}{\sqrt{1 - s^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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