Definition of the Derivative of a Function
The definition of the derivative in calculus, as the limit of the difference quotient, is explored interactively using an applet.
A video that explains the concept of the derivative of a Function is available.
The function to be explored has the form f(x) = a sin (b x + c) + d where a, b, c and d are parameters that can be changed. A secant through two points P and Q (on the graph of function f) is made to approach what is called the tangent line, at point P, to the graph of f.
1 - click on the button above "click here to start" and MAXIMIZE the window obtained.
2 - We start with two points $P$ and $Q$ on the graph of function f(x)= sin(x) (a = 1, b = 1, c = 0 and d = 0). Click on the button to make h (the difference in the x-coordinates of P and Q) smaller and notice how the secant through P and Q approaches what we call a tangent line to the graph of f. An important behavior to observe is also the convergence of the slope of the secant PQ to a finite value and that is the slope of the tangent line at x0.
3 - Use the slider to change the coordinate of P and repeat the above experiment. Each time note the geometrical behavior of the secant and the convergence of its slope to a finite value which is called the derivative at that point.
4 - Use the sliders to change a, b, c and d. Each time note the geometrical behavior of the secant and the convergence of its slope to a finite value which is called the derivative at that point.
More references on derivatives and differentiation.