`y = 1/2 (xsqrt(4-x^2) + 4arcsin(x/2))` Find the derivative of the function

The derivative of y in terms of x is denoted by  `d/(dx)y` or `y'` .

 For the given problem: `y =1/2[xsqrt(4-x^2)+4arcsin(x/2)]` , we apply the basic derivative property:

`d/(dx) c*f(x) = c d/(dx) f(x)` .

Then,

`d/(dx)y =d/(dx) 1/2[xsqrt(4-x^2)+4arcsin(x/2)]`

`y’ =1/2 d/(dx) [xsqrt(4-x^2)+4arcsin(x/2)]`

Apply the basic differentiation property: `d/(dx) (u+v) = d/(dx) (u) + d/(dx) (v)`

`y’ =1/2[d/(dx) (xsqrt(4-x^2))+ d/(dx) (4arcsin(x/2))]`

For the derivative of `d/(dx) (xsqrt(4-x^2))` , we apply the Product Rule: `d/(dx)(u*v) = u’*v =+u*v’` .

`d/(dx) (xsqrt(4-x^2))= d/(dx)(x) *sqrt(4-x^2)+ x * d/(dx) (sqrt(4-x^2))`

Let `u=x` then `u'= 1`

    `v= sqrt(4-x^2) ` then `v' =-x/ sqrt(4-x^2)`

Note: `d/(dx) sqrt(4-x^2) = d/(dx)(4-x^2)^(1/2)`

Applying the chain rule of derivative:

`d/(dx)(4-x^2)^(1/2)= 1/2(4-x^2)^(-1/2)*(-2x)`

                     ` =-x(4-x^2)^(-1/2)`

                    `=-x/(4-x^2)^(1/2)`  or - `–x/sqrt(4-x^2)`

 Following the Product Rule, we set-up the derivative as:

`d/(dx)(x) *sqrt(4-x^2)+ x * d/(dx) (sqrt(4-x^2))`

`= 1 * sqrt(4-x^2)+ x*(-x/sqrt(4-x^2))`

`= sqrt(4-x^2)-x^2/sqrt(4-x^2)`

 Express as one fraction:

`sqrt(4-x^2)* sqrt(4-x^2)/ sqrt(4-x^2)-x^2/sqrt(4-x^2)`

`=( sqrt(4-x^2))^2/ sqrt(4-x^2) –x^2/sqrt(4-x^2)`

`=( 4-x^2)/ sqrt(4-x^2) –x^2/sqrt(4-x^2)`

`=( 4-x^2-x^2)/ sqrt(4-x^2)`

`=( 4-2x^2)/ sqrt(4-x^2)`

Then, `d/(dx) (xsqrt(4-x^2))= ( 4-2x^2)/ sqrt(4-x^2)`

For the derivative of `d/(dx) (4arcsin(x/2))` , we apply the basic derivative property: `d/(dx) c*f(x) = c d/(dx) f(x)` .

`d/(dx) (4arcsin(x/2))= 4 d/(dx) (arcsin(x/2))`

Apply the basic derivative formula for inverse sine function: `d/(dx) (arcsin(u))= (du)/sqrt(1-u^2)` .

Let `u =x/2` then `du=1/2`

`4d/(dx) (4arcsin(x/2))]= 4*(1/2)/sqrt(1-(x/2)^2)`

                    `= 2/sqrt(1-(x^2/4))`

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

                     ` = 2/sqrt((4-x^2)/4)`

                    ` = 2/ (sqrt(4-x^2)/sqrt(4))`

                   `=2/ (sqrt(4-x^2)/2)`

                   `=2*2/sqrt(4-x^2)`

                   `=4/sqrt(4-x^2)`

Combining the results, we get:

`y' = 1/2[d/(dx) (xsqrt(4-x^2))+ d/(dx) (4arcsin(x/2))]`

`=1/2[( 4-2x^2)/ sqrt(4-x^2)+4/sqrt(4-x^2)]`

`=1/2[( 4-2x^2+4)/ sqrt(4-x^2)]`

` =1/2[( -2x^2+8)/ sqrt(4-x^2)]`

` =1/2[( 2(-x^2+4))/ sqrt(4-x^2)]`

` =(-x^2+4)/ sqrt(4-x^2)]`

or `y'=(4-x^2)/ sqrt(4-x^2)]`

Applying Law of Exponents:  ` x^n/x^m= x^n-m` :

`y' =(4-x^2)/ sqrt(4-x^2)`

` =(4-x^2)^1/ (4-x^2)^(1/2)`

` =(4-x^2)^(1-1/2)`

`=(4-x^2)^(1/2)`

Final answer:

`y'=(4-x^2)^(1/2)`

 or

`y'=sqrt(4-x^2)`