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
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
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.