Question

How do you find the instantaneous rate of change of y with respect to x at an arbitrary value of `X_o` if `y=2x^2` and `X_o=0`?

Answer 1

If `y=f(x)=2x^2`, then by the power rule, the instantaneous rate of change of `y` with respect to `x` is the derivative `dy/dx=f'(x)=4x`.

At an arbitrary point `x=X_{0}`, this derivative is just `f'(X_{0})=4X_{0}`. If `X_{0}=0`, then `f'(X_{0})=f'(0)=4\cdot 0=0`.