Let `G(x)= 2sin^(-1)((sqrt(x))/2)` Find G'(x)Please show every single step, without using any shortcuts. Thank you.

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lemjay | High School Teacher | (Level 2) Senior Educator

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To determine the derivative of inversine, use the formula:

`d/(du)sin^(-1)u= 1/sqrt(1-u^2)*u'`

Then let,


To determine u', apply the power formula of derivatives which is `d/(dx)cx^n=c*nx^(n-1)` .


Then, substitute u and u' to the formula of the derivative of inverse sine.






To simplify further, use the rule of radicals which is `root(n)(a)*root(n)(c)=root(n)(a*c)` .


Hence, the derivative of `G(x)=2sin^(-1)(sqrtx/2)` is `G'(x)=1/sqrt(x(4-x))` .

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