DERIVATIVES II

Derivative of a function at a point

Tangent to a curve at a point

As has already been seen the tangent to a point of a curve is obtained as the limit of the secants at this point.  

1.- Check how as h tends towards zero, that is to say, as the point Q approaches P, the secant QP increasingly approaches the tangent.  

The slopes of the secants:

All the secants pass through the point P (a, f(a)) and through the point Q (a+h, f(a+h)). Therefore the slope of the secants are:

2.- Watch how the slopes of the secants vary when the point Q approaches the point P.  

a) Calculate the slope of the tangent straight line at the point of abscissa 1.  

b) Calculate the slope at other points x=2; x=2; x=0; x=-1, etc.

c) Write the equation of the straight line tangent to the curve of the figure at the point x=1.

d) Write the equation of the straight line tangents at the points where you have calculated the slopes.

Definition of the derivative of a function at a point

The derivative of a function at the point of abscissa x = a is the slope of the tangent to the curve, which represents that function, at the point (a,f(a)).

3.- Observe and note the derivative at different points: x=1; x=2; x=0; x=-1, etc.

a) Find two points with a derivaive of zero.

b) Find two points with derivative  2; 10; -2; -10; etc.

c) Observe how in each case the slopes of the secants QP approach the derivative. 

Calculation of the derivative of a function at a point

Let y = f(x) be a function.

The derivative of f(x) at the point x=a, as we have seen, is the slope of the straight line tangent to the curve at the point P(a,f(a)) and is designated by f ' (a).

We have seen that:

Furthermore, the slope of the secants, for each value of h, are:

Therefore, as the slopes of the secants increasingly approach the slope of the tangent, we can write:

4.- Check again how the values of m approach the derivative when h tends towards zero in the following points:

a) In x = 1.5.

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

Use the change of scale and the diplacement of the axes when you need to see the graphs with more detail. 

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000