Let `G(x) = 2sin^(-1)((sqrt(x))/(2))` 1) Find G'(x). 2) The domain of G(x)is: 3) The domain of G'(x) is:  

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

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(1) To determine G'(x), use the formula `d/(du) sin^(-1)u= 1/sqrt(1-u^2)*u'` .

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




>> Hence, `G'(x) =1/(sqrtxsqrt(4-x)) ` .

(2) To determine the domain of G(x), we need to take note the domain of the basic inverse sine function.

For `y=sin^(-1) x` , domain is `-1lt=xlt=1` .

So the domain of `G(x) = sin^(-1)(sqrtx/2)`  should satisfy the condition `-1lt=sqrtx/2lt=1` .

To solve for x, set

`sqrtx/2 >=-1`                  and                  `sqrtx/2lt=1`

`sqrtxgt=-2`                                        `sqrtxlt=2`                

`xlt=4 `                                               `xlt=4`  

Also for square roots, the radicand shouls be positive So,the values of x in `sqrtx` should be  `x>=0` .

>> Hence, the domain of G(x) is `0<=x<=4` .

(3) To determine the domain of G'(x), we need to take note that in rational functions, we can not have a zero denominator.

Moreover, the denominator of G'(x) contains square roots. In square root, a negative radicand is not allowed.

Since we cannot have a zero denominator and a negative radicand, set each radicand in the denominator greater than zero.

 `4-x>0`                    and                 `x > 0`

   `-x > -4`

      `x < 4`

>> Hence, the domain of G'(x) is `0ltxlt4` .

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