The function y=f(x) in green, is associated with two lines in the above diagram. The line there in blue cuts the function at two points while the line in orange touches the curve at a single point. With some of our previus knowledge, we would say that the line in orange is the tangent line as it touches the curve at one single point. The statement IS ture, but does it mean that the tangent cannot cut the curve furthur at another point?
Well, the figure below answers the question.
Here, we see that the tangents at P & Q do intersect the curve of the sine function again at other points. We also notice that the curve has the same nature between point P & the point 3pi/2 & the tangent intersects the curve on the left of 3pi/2, ie when the curve has changed its nature! We find the same observation for the tangent at Q. So, for any curve, the tangent may intersect it again at a point provided the curve has changed its nature atleast once between the new point of intersection & the point of contact of the tangent.
So, that was about the orange(tangent) line. Lets talk about the blue line. Have you seen something like this before? Sure you have. Have a look at the image below.
So, this line in red on the circle, and in blue on the function, is called the secant line. Now, lets try to find ut the slope of this secant line.
So, the slope comes out to be (the change in y)/(the change in x) & here, the change in y refers to the difference between the value of the function at A and its value at B, which is:
Hence, The slope of the secant line can be written as:
Now, we need to get the orange line into the picture, keeping the blue line in mind.
Now, Lets take our observation a step furthur. You can observe in the figure below, that when the above happens, ie when A gets closer to B, the change in x goes on decreasing and finally, tends to zero when the slopes tend to be equal, or when A & B tend to coincide.
So, we basically have limiting values here which calls for the application of limits.
And that limiting value there folks, is the derivative!