INCREASING AND DECREASING FUNCTIONS  
Analysis  
One of the first applications of the derivative can be found by studying how a function increases and decreases.
You should already be familiar with the graphs of increasing and decreasing functions.
1. THE RELATION BETWEEN THE DERIVATIVE AND WHETHER THE FUNCTION INCREASES OR DECREASES 

Let f be a function with a derivative. We can say that a function y=f(x) INCREASES at x_{o} when there is a neighbourhood of x_{o }such that:


If f is a function with a derivative then:


We can say that a function y=f(x) DECREASES at x_{o} when there is a neighbourhood of x_{o }such that:


In this case:


Look carefully at the graph of the function y=f(x) and its derivative y=f'(x) in the window. 

Remember that the derivative of a function at a point is the same as the gradient of the tangent to the curve of the function at that point.

María José García Cebrian  
Spanish Ministry of Education. Year 2001  
Except where otherwise noted, this work is licensed under a Creative Common License