Given an implicit function
$$ f(t,x) = 0 $$ and a solution pair $(0,x_0)$. I would like to compute $(1,x_1)$ from the known solution by solving the corresponding ODE given by the total derivative of $f$ $$ \frac{d x}{d t} = -\frac{\partial f / \partial t}{\partial f / \partial x}. $$
I understand that I can use a predictor-corrector method to solve this ODE: predictor with any ODE scheme and corrector with Newton-method to enforce the step to be a solution of $f$.
I have difficulties to find a proper reference on this approach and its theoretical background. Also, is there an ODE solver which allows a corrector step (preferably in MATLAB)?