Question

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

Answer 1

It is `dy/(ds)=-(2s^2)/(sqrt(1-s^2))`

Explanation:

It is `y(s)=s*sqrt(1-s^2) + cos^(-1)(s)` hence its derivative 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

The derivative of `cos^(-1)(s)=arccos(s)` is `-(1/(sqrt(1-s^2)))`