The function $f:[0,1]\to \mathbb{R}$ is continuous, $f(0)<0$, $f(1)>0$ and there is one root in between. Using $f(0)$ and $f(1)$, the expression $\frac{1\cdot f(0)-0\cdot f(1)}{f(0)-f(1)}$ would approximate the root.
Question is, if $f'(0)$ and $f'(1)$ are available too, what would an expression look like? I have been trying to solve and simplify some cubic equations with no success, and I think the problem might be simpler.
If you write a cubic equation you then have to find the solution of that equation... which is not very simple.
Instead I would suggest to take the two tangent lines in the points $(0,f(0))$ and $(1,f(1))$, find the two intersection of these lines and take the mean value of them, if both points are inside the interval. Otherwise if only one point is inside the interval take that point. If both points are outside, fall back to your original method.
But see also Newton's method.