Calculus of a Single Variable, Chapter 5, 5.6, Section 5.6, Problem 56

y=25arcsin(x/5) - xsqrt(25-x^2)
Before taking the derivative, express the radical in exponent form.
y=25arcsin(x/5) - x(25-x^2)^(1/2)
To get y', take the derivative of each term.
y' = d/dx[25arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]
y' = 25d/dx[arcsin(x/5)] - d/dx[x(25-x^2)^(1/2)]
Take note that the derivative formula of arcsine is d/dx[arcsin(u)] = 1/sqrt(1-u^2)*(du)/dx .
Applying that formula, y' will become:
y'=25* 1/sqrt(1-(x/5)^2) *d/dx(x/5) - d/dx[x(25-x^2)^(1/2)]
y'=25* 1/sqrt(1-(x/5)^2) *1/5 - d/dx[x(25-x^2)^(1/2)]
y'=25* 1/sqrt(1- x^2/25)*1/5 - d/dx[x(25-x^2)^(1/2)]
y'=25* 1/((1/5)sqrt(25- x^2))*1/5 - d/dx[x(25-x^2)^(1/2)]
y'=25/sqrt(25-x^2) -d/dx[x(25-x^2)^(1/2)]
To take the derivative of the second term, apply the product rule d/dx(u*v) = u*(dv)/dx + v*(du)/dx .
Applying this, the y' will be:
y'=25/sqrt(25-x^2) - [x*d/dx((25-x^2)^(1/2)) + (25-x^2)^(1/2)*d/dx(x)]
Also, use the derivative formula d/dx(u^n) = n*u^(n-1)*(du)/dx .
y'=25/sqrt(25-x^2) - [x*1/2*(25-x^2)^(-1/2)*d/dx(25-x^2) + (25-x^2)^(1/2)*1]
y'=25/sqrt(25-x^2) - [x*1/2*(25-x^2)^(-1/2)*(-2x) + (25-x^2)^(1/2)*1]
y'=25/sqrt(25-x^2) - [-x^2(25-x^2)^(-1/2) + (25-x^2)^(1/2)]
Then, express this with positive exponent only.
y'=25/sqrt(25-x^2) - [-x^2/(25-x^2)^(1/2) + (25-x^2)^(1/2)]
Also, convert the fractional exponent to radical form.
y'=25/sqrt(25-x^2) - [-x^2/sqrt(25-x^2) + sqrt(25-x^2)]
So the derivative of the function simplifies to:
y'=25/sqrt(25-x^2) +x^2/sqrt(25-x^2) - sqrt(25-x^2)
y'= (x^2+25)/sqrt(25 - x^2) - sqrt(25-x^2)
y'= (x^2+25)/sqrt(25 - x^2) - sqrt(25-x^2)/1* sqrt(25-x^2)/sqrt(25-x^2)
y'= (x^2+25)/sqrt(25 - x^2)-(25-x^2)/sqrt(25-x^2)
y'= (x^2+25 - (25-x^2))/sqrt(25-x^2)
y'=(2x^2)/sqrt(25-x^2)

Therefore, the derivative of the function is y'=(2x^2)/sqrt(25-x^2) .