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

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Answer 1 · 2014-09-21T13:26:04

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 ${dy}/{dt}=6$ ,

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

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