We need to differentiate this implicitly, because we don't have a function in terms of one variable.
When we differentiate #y# we use the chain rule:
#d/dy*dy/dx=d/dx#
As an example if we had:
#y^2#
This would be:
#d/dy(y^2)*dy/dx=2ydy/dx#
In this example we also need to use the product rule on the term #xy^2#
Writing #sqrt(y)# as #y^(1/2)#
#y^(1/2)+xy^2=5#
Differentiating:
#1/2y^(-1/2)*dy/dx+x*2ydy/dx+y^2=0#
#1/2y^(-1/2)*dy/dx+x*2ydy/dx=-y^2#
Factor out #dy/dx#:
#dy/dx(1/2y^(-1/2)+2xy)=-y^2#
Divide by #(1/2y^(-1/2)+2xy)#
#dy/dx=(-y^2)/((1/2y^(-1/2)+2xy))=(-y^2)/(1/(2sqrt(y))+2xy#
Simplify:
Multiply by: #2sqrt(y)#
#(-y^2*2sqrt(y))/(2sqrt(y)1/(2sqrt(y))+2xy*2sqrt(y)#
#(-y^2*2sqrt(y))/(cancel(2sqrt(y))1/(cancel(2sqrt(y)))+2xy*2sqrt(y)#
#(-y^2*2sqrt(y))/(1+2xy*2sqrt(y))=-(2sqrt(y^5))/(1+4xsqrt(y^3))=color(blue)(-(2y^(5/2))/(1+4xy^(3/2)))#
