We start with a point `Q(1, 1)` which is somewhere near `P(2,4)`:

Graph tangent at point first principle PQ 1

The slope of PQ is given by:

`m=(y_2-y_1)/(x_2-x_1)`

`=(4-1)/(2-1)`

`=3`

Now we move Q further around the curve so it is closer to P. Let's use `Q(1.5,2.25)` which is closer to `P(2,4)`:

Graph tangent at point first principle PQ

The slope of PQ is now given by:

`m=(y_2-y_1)/(x_2-x_1)`

`=(4-2.25)/(2-1.5)`

`=3.5`

We see that this is already a pretty good approximation to the tangent at P, but not good enough.

Now we move Q even closer to P, say `Q(1.9,3.61)`.

Now we have:

Graph tangent at point first principle slope 1.9


So

`m=(y_2-y_1)/(x_2-x_1)`

`=(4-3.61)/(2-1.9)`

`=3.9`

We can see that we are very close to the required slope.

Now if Q is moved to `(1.99,3.9601)`, then slope PQ is `3.99`.

If Q is `(1.999,3.996001)`, then the slope is `3.999`.

Clearly, as `x → 2`, the slope of `PQ → 4`. But notice that we cannot actually let `x = 2`, since the fraction for m would have `0` on the bottom, and so it would be undefined.

We have found that the rate of change of y with respect to x is `4` units at the point `x = 2` .