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 3) Senior Educator

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`G(x)=2sin^(-1)(sqrtx/2)`

`G'(x)=2d/(dx)sin^(-1)(sqrtx/2)`

To determine the derivative of inversine, use the formula:

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

Then let,

`u=sqrtx/2=1/2x^(1/2)`

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

`u'=1/2*1/2x^(-1/2)=1/(4sqrtx)`

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

`G'(x)=2*1/sqrt(1-(sqrtx/2)^2)*1/(4sqrtx)`

`G'(x)=1/(2sqrtxsqrt(1-x/4))`

`G'(x)=1/(2sqrtxsqrt((4-x)/4))`

`G'(x)=1/(2sqrtx*1/2sqrt(4-x))`

`G'(x)=1/(sqrtxsqrt(4-x))`

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

`G'(x)=1/sqrt(x(4-x))`

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

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