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.
First Derivative of Function
Solve Tangent Lines Problems in Calculus
Solve Rate of Change Problems in Calculus
Rules of Differentiation of Functions in Calculus
Differentiation of Trigonometric Functions
Use the Chain Rule of Differentiation in Calculus