Question

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

Answer 1

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`