The derivative of a function f(x) at an x-coordinate point x = a is:
the gradient of the tangent
to the curve, represented by this function, at point P (a,f(a)).

1.- Look carefully and write down the derivative of these points:x=1; x=2; x=0; x=-1, etc.

2.- Find two points whose derivatives are zero.

3.- Find the points with derivatives 2; 5; 10; -2; -7; etc.

4.- Note that for each point you choose, the gradients of the secants OP approach the derivative.

Let the function be y = f(x). The derivative of f(x) at point x=a, as we have already seen, is the gradient of the tangent to the curve at point P(a,f(a)) and can be written as f ' (a).

We have already seen that the tangent is the limit of secants QP when Q approaches P:

Furthermore, the gradients of the secants, for each value of h, are equal to:

Therefore, we can define the derivative as the limit of the gradients of the secants when Q approaches P, i.e. when h tends to zero:

5.- Show once again how the values of m approach the derivative when h tends to zero at the following points:

a) At x = 1.5.

b) At x = 0; x = -1; x = -2; etc.

6.- Find the equation of the tangent for each point in the activity above.