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.
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:
Then to find the rate of change with respect to x we take the derivative:
`(dd)/(dx)=1/2(x^4+3x^2+1)^(-1/2)(4x^3+6x)` using the chain rule
At x=-1 we have `(dd)/(dx)=(2(-1)^3+3(-1))/sqrt((-1)^4+3(-1)^2+1)`