Learning Objectives
A student will be able to:
 Demonstrate an understanding of the derivative of a function as a slope of the tangent line.
 Demonstrate an understanding of the derivative as an instantaneous rate of change.
 Understand the relationship between continuity and differentiability.
The function
The Derivative
The function
where
Based on the discussion in previous section, the derivative
Example 1:
Find the derivative of
Solution:
We begin with the definition of the derivative,
where
Substituting into the derivative formula,
Example 2:
Find the derivative of
Solution:
Using the definition of the derivative,
Thus the slope of the tangent line at
For
Thus the equation of the tangent line is
Notation
Calculus, just like all branches of mathematics, is rich with notation. There are many ways to denote the derivative of a function
In addition, when substituting the point
Existence and Differentiability of a Function
If, at the point
if
then the derivative
 At a corner. For example
f(x)=x , where the derivative on both sides ofx=0 differ (Figure 4).  At a cusp. For example
f(x)=x2/3 , where the slopes of the secant lines approach+∞ on the right and−∞ on the left (Figure 5).  A vertical tangent. For example
f(x)=x1/3 , where the slopes of the secant lines approach+∞ on the right and−∞ on the left (Figure 6).  A jump discontinuity. For example, the step function (Figure 7)
where the limit from the left is
Figure 4
Figure 5
Figure 6
Figure 7
Many functions in mathematics do not have corners, cusps, vertical tangents, or jump discontinuities. We call them differentiable functions.
From what we have learned already about differentiability, it will not be difficult to show that continuity is an important condition for differentiability. The following theorem is one of the most important theorems in calculus:
Differentiability and Continuity
If
The logically equivalent statement is quite useful: If
(The converse is not necessarily true.)
We have already seen that the converse is not true in some cases. The function can have a cusp, a corner, or a vertical tangent and still be continuous, but it is not differentiable.
Multimedia Links
For an introduction to the derivative (4.0)(4.1), see Math Video Tutorials by James Sousa, Introduction to the Derivative (9:57).
The following simulator traces the instantaneous slope of a curve and graphs a qualitative form of derivative function on an axis below the curve Surfing the Derivative.
The following applet allows you to explore the relationship between a function and its derivative on a graph. Notice that as you move x along the curve, the slope of the tangent line to
For a video presentation of differentiability and continuity (4.3), see Differentiability and Continuity (6:34).
Review Questions
In problems 1 through 6, use the definition of the derivative to find

f(x)=6x2;x0=3 
f(x)=x+2−−−−√;x0=8 
f(x)=3x3−2;x0=−1 
f(x)=1x+2;x0=−1 
f(x)=ax2−b, (wherea andb are constants);x0=b 
f(x)=x1/3;x0=1 .  Find
dy/dxx=1 given thaty=5x2−2.  Show that
f(x)=x√3 is defined atx=0 but it is not differentiable atx=0 . Sketch the graph.  Show that is continuous and differentiable at
f(x)={x2+12xx≥1x>1 x=1 . Hint: Take the limit from both sides. Sketch the graph off .
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Date Created:
Feb 23, 2012Last Modified:
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