Two points determine a line. We select one point on the graph of y = f(x) by specifying a domain value a; the corresponding point on the graph has coordinates (a,f(a)). We can select another nearby point on the graph by choosing a (small) value of h. This nearby point has coordinates (a+h,f(a+h)). The line joining these two points is a secant line to the graph of f.

One Secant

Slope of a Secant Line
slope =
f(a+h) - f(a)


The slope of a secant line is a difference quotient!!

Each nonzero choice of h determines a point (a+h,f(a+h)) and the secant line that joins this point to (a,f(a)). An interesting thing happens when we choose smaller and smaller values of h. We have prepared a crude animation in order to illustrate. It's so crude that you must advance the frames yourself. Just push the button and watch what happens to the secant lines!!

If f is differentiable at a, then as h-->0, the secant lines gather at a limiting line, which is the tangent line to the graph at (a,f(a)). Since the slopes of the secant lines are difference quotients, the slope of the tangent line is the limit of the difference quotients as h approaches 0...the slope of the tangent line is f '(a)!!! As promised!!

The tangent line to y = f(x) at (a,f(a))
Tangent Line

What's Next: We have one more thing to do before we take up the study of differentiation techniques (i.e. how to compute the derivative of a function). We investigate the relationship between differentiability and continuity.



© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson