Let y = f(x) be a function, and let P(a, f(a)) and Q(a+h, f(a+h)) be two points on the graph of the function that are close to each other. This graph is shown below:

Joining the points P and Q with a straight line gives us the secant on the graph of the function. And in the the gradient of the line is given by

m = =

In limiting process, i.e. as Q approaches P, h becomes really small, almost close to zero. So

m =

= =

=

As , the chord/secant PQ tends to be a tangent at P for the curve y = f(x). Thus the limiting becomes the slope of the tangent at P for y = f(x).

We denote = , and

= f'(a)

Therefore = f'(a) = m

m as we’ve seen is the slope or gradient of the tangent at P for f(x).

f'(a) = tan = slope of f(x) at x = a.