derivative of ((Sqrt(x+3y)+sqrt(3xy))=(sqrt(21)+sqrt(90)) at point (6,5)google

Answer 1

Check that (6,5) lies on the curve by substituting x=6,y=5 and see if the equality holds. (yes, it holds).

Differentiate:
((Sqrt(x+3y)+sqrt(3xy))=(sqrt(21)+sqrt(90))
(1/2)/sqrt(x+3y)*(1+3y')+(1/2)/sqrt(3xy)*(3y+3xy')=0
transpose and cross multiply:
-sqrt(x+3y)*(1+3y')=sqrt(3xy)*(3y+3xy')
Solve for y'

y'(x,y)=-(sqrt(3*y+x)+3^(3/2)*y*sqrt(x*y))/(3*sqrt(3*y+x)+3^(3/2)*x*sqrt(x*y))

so
y'(6,5)=(-5*3^(3/2)*sqrt(30)-sqrt(21))/(2*3^(5/2)*sqrt(30)+3*sqrt(21))
=-0.796....

please check my arithmetic.