I want to ask a question about linking gravitational field strength g and gravitational potential V.
If I consider both quantities at a distance r from the centre of planet of mass M then I can see that they are closely related:
if $g = \frac{GM}{r^2}$ and $V = -\frac{GM}{r}$ then at a given point,
$$g = -\frac{-GM}{r}\times\frac{1}{r} = -\frac{V}{r}$$
so $$g = -\frac{V}{r}$$
Now, the book I am reading also states that I can link these variables through the graphs.
The rate of increase of the gravitational potential with distance r is known as the potential gradient, $\frac{\Delta V}{\Delta r}$ and that this value is equal to g, the gravitational field strength at that distance.
I see this is possible because using the equation we derived previously:
$$g = -\frac{V}{r}$$ $$-gr = V$$ $$-g = \frac{\Delta V}{\Delta R}$$ $$g = -\frac{\Delta V}{\Delta R}$$
but my question is this: using the graph, why can I not just take a point on the line and use the formula $g = -\frac{V}{r}$ instead of differentiation using $g = -\frac{\Delta V}{\Delta R}$ to find the value of g at a distance r?