Consider the graph below, where A is ((x, x+h)) and B is ((f(x), f(x+h))).

mtaimg/images/topics/7/7-70-1.png/mtaimg

The gradient of the line segment (AB) is (\dfrac{f(x+h) - f(x)}{x+h - x} \implies \dfrac{f(x+h) - f(x)}{h} ).

As the value of (h) decreases, the gradient of (AB) approaches the gradient (f'(x)) of the tangent at (A).

This can be expressed as a blimit/b:

(\boxed{f'(x) = \lim\limits_h→0 \Big(\dfrac{f(x+h) - f(x)}{h} \Big)})

This is known as bdifferentiation from first principles/b.

[b]uDifferentiation from first principles[/u]/b

(f'(x) = \lim\limits_h→0 \Big(\dfrac{f(x+h) - f(x)}{h} \Big))