The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If  is positive, then must be increasing. If is negative, then  must be decreasing. If is zero, then must be at a relative maximum or relative minimum.  tells us similar things about .  also gives us valuable information about . In particular it tells us when the function is concave up, concave down, or there is a point of inflection. This same type of information is indicated about  by  and so on.
 

increasing	+
decreasing	-
relative min. or max.	0
concave up	increasing	+
concave down	decreasing	-
point of inflection	relative min. or max.	0
	concave up	increasing	+
	concave down	decreasing	-
	point of inflection	relative min. or max.	0
		concave up	increasing
		concave down	decreasing
		point of inflection	relative min. or max.
			concave up
			concave down
			point of inflection

	The following Maple command results in the graph at the left: > plot([x^3+3*x^2,3*x^2+6*x],x=-4..4,y=-10...10); The red graph is the graph of the function:and the green graph is the graph of its derivative:
Find the interval(s) on the function where the function is increasing.	Did you get  and  ?
Now take a look at the green derivative curve over the same intervals. Do you notice anything?	Do you see that over these same intervals, that the derivative curve is above the x-axis? Note: a positive number on the derivative curve correlates with the original function while it is increasing.
Where on curve of the red function do you spot that the curve is changing from increasing to decreasing? (These points are called critical points)	Did you answer when x = -2 and when x = 0 ?
Now take a look at the green derivative curve at those two values of x. Do you notice anything?	Do you see that at these two points, that the derivative curve is equal to zero? Note: when the derivative curve is equal to zero, the original function must be at a critical point, that is, the curve is changing from increasing to decreasing or visa versa.
Find the interval(s) on the function where the function is decreasing.	Did you get   ?
Now take a look at the green derivative curve over the same interval. Do you notice anything?	Do you see that over this same interval, that the derivative curve is below the x-axis? Note: a negative number on the derivative curve correlates with the original function while it is decreasing.

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 2.10. This module will allow you to practice using graphical information about f ' to determine the slope of the graph of f..

Definition:

Here we have the reverse of the process that we have been studying. We begin with the derivative, and we want to find the function. This type of discovery process is common to scientific experimentation and data gathering.

First, we need to aknowledge that different functions can result in the exact same derivative. Look at the example below:
 
 

	Here we see a family of curves plotted with their common derivative. The family of parabolic functions is: , where c takes on the values: -1, 0, 1, 2, 3, and 4.
The straight line in the graph above is . It is the  derivative function for all six of the parabolic functions.	Because a derivative is primarily a tool for determining the shape of a function, the position of a graph does not affect the shape. Therefore congruent curves that are oriented the same, but have a different position have the same derivative.

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