If ##x^2+y^2=25## and ##dy/dt=6##, how do you find ##dx/dt## when ##y=4## ?

There are two values depending on the point.

##{dx}/{dt}={(-8 " at " (3,4)),(8 " at "(-3,4)):}##

Let us look at some details.

First, let us find the values of ##x##.

By plugging in ##y=4## into ##x^2+y^2=25##,

##x^2+16=25 Rightarrow x^2=9 Rightarrow x=pm3##

Now, let us find some derivatives.

By differentiating with respect to ##t##,

##d/{dt}(x^2+y^2)=d/{dt}(25) Rightarrow 2x{dx}/{dt}+2y{dy}/{dt}=0##

by dividing by ##2x##,

##Rightarrow {dx}/{dx}+y/x{dy}/{dt}=0##

by subtracting ##y/x{dy}/{dt}##,

##Rightarrow {dx}/{dt}=-y/x{dy}/{dt}##

Since ##y=4##, ##x=pm3##, and ##{dy}/{dt}=6##,

##{dx}/{dt}=-{4}/{pm3}(6)=pm8##