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The Sign of the Derivative

Increasing, Decreasing, Stationary

Let f(x) be a function and assume that for each value of x, we can calculate the slope of the tangent to the graph y = f(x)  at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f(x) and denote it by f ´ (x).

 

   

The derivative of f(x) at the point x is equal to the slope of the tangent
to y = f(x) at x.

   

The graph of a function y = f(x) in an interval is increasing (or rising) if all of its tangents have positive slopes. That is, it is increasing if as x increases, y also increases.


The graph of a function y = f(x) in an interval is decreasing (or falling) if all of its tangents have negative slopes. That is, it is decreasing if as x increases, y decreases.


The graph of a function y = f(x) has a stationary point at the point where the tangent is horizontal or has zero slope. This always occurs at the points where a function changes from increasing to decreasing and at the points where a function changes from decreasing to increasing. It can also occur at other points and we will discuss this possibility later.


When the graph of a function y = f(x) is vertical or discontinuous the tangent is undefined.

These basic properties of the derivative are summarized in the following table.

Behaviour Graphs

Derivative
(slope of tangent)

Graph increasing f ´(x) > 0
(positive)
Graph decreasing f ´(x) < 0
(negative)
Tangent horizontal f ´(x) = 0
(zero)
Tangent vertical
or No Tangent
f ´(x) undefined


Example

Let . Then .
Since f'(x)=cosx>0 over the intervals (-π/2, π/2), (3π/2, 5π/2), and (7π/2, 9π/2), the function is increasing over those intervals.
Asf'(x)=cosx<0 over the intervals (-3π/2, -π/2), (π/2, 3π/2), and (5π/2, 7π/2) the function is decreasing over those intervals.

          

 

Exercise 1

Test that the properties stated in the above table are true. You can examine the fourteen examples provided in the scroll bar on the top of the applet below or enter your own function in the box provided. If you enter your own function, you must use the symbols + for add, for subtract, * for multiply,  / for divide, and ^ to raise to a power. You can also use various mathematical functions: sin, cos, tan, sec, cot, csc, arcsin, arccos, arctan, exp, ln, log2, log10, abs, sqrt and cubert. (Here, "abs" is the absolute value function, "sqrt" is the square root function and "cubert" is the cube root function.)

Make sure you understand the following connections between the two graphs.
  • When the graph of the function y= f(x) is horizontal then the graph of its derivative y= f '(x) passes through the x axis (is equal to zero). This occurs only at a stationary point.
  • When the slope of the function y= f(x) is positive, the graph of its derivative y= f '(x) is above the x-axis (is positive).
  • When the slope of the function y= f(x) is negative, the graph of its derivative y= f '(x) is below the x-axis (is negative).
  • When the slope of the function y= f(x) is vertical , the graph of its derivative y= f '(x) is undefined at that value of x.

Can't see the above java applet? Click here to see how to enable Java on your web browser. (This applet is based on free Java applets from JavaMath )


Exercise 2A

Consider the function f(x) whose graph is shown below.

graph that is increasing between 0 and 1 and decreasing between 1 and 2 and increasing for x greater than 2.

When x < 1, f(x) is increasing decreasing
When 1< x < 2, f(x) is increasing decreasing
When x > 2, f(x) is increasing decreasing


Click on the graph below that is f´(x), the derivative of f(x).

a(a)   b(b) c(c) d(d)

Example

Consider the rational function f(x) = 1 over (x - 4) . Its graph is a hyperbola with asymptote at x = 4. It is discontinuous at x = 4.

graph

Differentiating, f '(x) = -1 over (x - 4) squared. When x = 4, f '(x) = 1/0, which is not defined. All other values of x have a negative derivative and an associated decreasing slope on the graph. The graph of the derivative is below.

graph of derivative

 

Exercise 2B

Consider the functions in these 8 exercises.

 

 

 


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