The derivative f '(x) of a function f(x) for some value of x, x0, is the instantaneous slope of the graph of the function at the point (x0, f(x0)).
We can find the derivative of f(x) at x = x0 using the limit definition of a derivative:
For a difference Δx in x, f '(x) equals the limit as Δx approaches 0 of
[f(x + Δx) - f(x)]/Δx
This demo shows a line which connects two points whose x coordinates differ by Δx. As Δx approaches 0, the slope of the line approximates the derivative of the function at the point (x0, f(x0)), represented here by a red dot.
1. Type in any linear function for f(x) and then try changing the size of Δx. Why is the accuracy of the approximation independent of Δx here?
2. Use |x| (type "abs(x)") for f(x), and set x0 equal to 0. Try a small, positive value for Δx and then try a small negative value. What problem arises?