What is the relationship of the slope of the tangent line at a point and the derivative of the function?

Let there be some change in $x$ by $h$. If we were to draw a line cutting through the initial and resultant point, then we would have a line passing through $(x, f(x))$ and $(x+h, f(x+h))$ as shown in the diagram above. (This line is the secant line, as some may recognize).

The derivative at a point is essentially the resultant change in $f(x)$ as a ratio of an infinitesimally small change in $x$. This means that the derivative at a point is $\frac{f(x+h) - f(x)}{(x+h)-x}$, for some infinitesimally small $h$. But this is exactly the gradient of the secant line shown above.

If we were to make $h$ very small to produce the "infinitesimally small change in $x$" (i.e. let $h \to 0$), then notice that the secant line above eventually becomes the tangent line at the point $(x, f(x))$. The derivative at that point then eventually becomes equal to the slope of the tangent line.