Calculus of a Single Variable, Chapter 5, 5.6, Section 5.6, Problem 52

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)