What is the graphical meaning of the first derivative of a function in calculus? A graphical interpretation of the derivative of a function based on the difference quotient is explored interactively using an applet.
We start with the definition of the first derivative of a function f.
f ' (x) = lim [ f(x+h) - f(x) ] / h as h tends to 0.
Using the applet below, you are able to select function f(x) from a list of 5 functions, observe the graph of f(x) (blue color), the graph of f(x+h) (red color) and that of [ f(x+h) - f(x) ] / h (black color). You are also able to make h close to 0, by steps, and observe the graph of [ f(x+h) - f(x) ] / h approaches the graph of a certain function close to the first derivative.
APPLET
TUTORIAL
1 - click on the button above "click here to start" and MAXIMIZE the window obtained.
2 - Select f(x) = x^{ 2} and decrease h. For what value(s) of h do you think the graph of [ f(x+h) - f(x) ] / h is close enough to the graph of the first derivative of f which is f ' (x) = 2x?
3 - Select f(x) = e^{ x} and decrease h. How is the graph of f positioned with respect to the graph of f(x) and f(x+h)? What do you think is the first derivative of f(x) = e^{ x}?
4 - Select f(x) = sin(x) and decrease h. For what value(s) of h do you think the graph of [ f(x+h) - f(x) ] / h is close enough to the graph of the first derivative of f which is f ' (x) = cos(x)? Compare this value of h to the one in step 1 above.
More references on derivatives and differentiation.