Find the rate of change of the distance between the origin and the moving point on the graph of the function y=x^2+1; x=-1I know I have to use the distance formula, but the x^2 is just throwing me...

Find the rate of change of the distance between the origin and the moving point on the graph of the function y=x^2+1; x=-1

I know I have to use the distance formula, but the x^2 is just throwing me off. Also there are three different values of x that I have to find, but I know I just have to subsitute those in for x after I get an equation.

embizze | High School Teacher | (Level 1) Educator Emeritus

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We want to find the rate of change of the distance from a point on the graph of `y=x^2+1` to the origin.

The distance `d` is given by `d=sqrt((x-0)^2+(y-0)^2)` or `d=sqrt(x^2+y^2)` . We can substitute `x^2+1` for `y` to get:

`d=sqrt(x^2+(x^2+1)^2)`

Then to find the rate of change with respect to x we take the derivative:

`d=sqrt(x^2+x^4+2x^2+1)`

`d=(x^4+3x^2+1)^(1/2)`

`(dd)/(dx)=1/2(x^4+3x^2+1)^(-1/2)(4x^3+6x)` using the chain rule

`=(2x^3+3x)/sqrt(x^4+3x^2+1)`

At x=-1 we have `(dd)/(dx)=(2(-1)^3+3(-1))/sqrt((-1)^4+3(-1)^2+1)`

`=(-5)/sqrt(5)=-sqrt(5)~~-2.24`