The Derivative

 

Introduction:

 

     This is the first portion of my on-line Calculus tutorial program.  In this first of two lessons, we will discuss one basic element of Calculus, the derivative.  Some of you may not have heard this word before, but I’m sure you have heard of a slope and rate of change; these all mean the same thing.  In this lesson, you will learn how to use the derivative to solve various problems, ranging in complexity.  Also, you will learn various techniques to differentiate -- taking the derivative -- of complex equations.

 

Part One: The Slope

 

The derivative is the same as the slope or rate of change.  Therefore, we will begin our study of the derivative by examining the slope of a curve. 

 

Examine the graph below:

 

**********Graph Not Available**********This is a U-Shaped Curve**********


Now you may be wondering: “how can we find the slope of such a curved graph?”  If you zoom in some more, what do you think would happen to the shape of the curve?

 

**********Graph Not Available**********A Straighter Graph After Zooming In**********


Now, don’t you see a straighter line?  We can use this to find the derivative of a point on the graph.  Let’s pick the points (9,81) and (10,100). 

The slope is equal to:                  (x1 , y1)            (x2    ,    y2)

 

    Slope =  y2-y1  = ∆y

                              x2-x1     ∆x

   

    Slope = 100-81 = 19

10-9                1

 

What the above figure implies that from x=9 to x=10, the rate of change is:

 

19 unit change in y value

1 unit change in x value

 

In fact, if you zoom in even more, to the point that the changes in the x and y range become infinitesimally small, you get an even more accurate calculation of the slope:

 

Slope[1] = dx

               dy

Below, this is evident:

 

**********Graph Not Available**********A STRAIGHT Graph Resulting from Zooming********** Say we take the slope of (9, 81) to (9.2, 84.64).

 


Slope = dx = 84.6-81 = 3.6 = 18

            dy      9.2-9      0.2      1

 

Notice that we went from having a slope of 19 above to now having a derivative of 18.  This means that our calculations are becoming more and more accurate.  In fact if we continue to zoom in, we could get the derivative at the point.  Another way of finding the derivative is by using the notion of the limit, as below:

 

y’ = lim     f(x+h) - f(x)

                                                             h→0             h

 

y prime indicates the derivative.  We can see that the changes in the domain become infinitesimally small ( lim h → 0 ); the definition of the derivative. 

 

Part Two: Techniques of Differentiation

 

     In your study of physics or courses in economics or statistics, you will have to differentiate a given equation.  In physics, for example, you may have to differentiate a velocity equation to find the acceleration at a particular time, t.  In economics, taking the derivative of a total cost equation will give you the marginal cost as a function of quantity produced.  All these and much more, are common in these fields, and many more examples exist in other professions.

 

One of the basic derivatives is that a single variable.

 

Example:

 

Differentiate:  f(x) = x2.

 

Solution:        f’(x) = x(2-1). 1  = 1 x.

                                          2     2

                               

·        In general, if you have an equation of the form:  

 

f(x) = xN,

 

     The derivative is:   

 

d[f(x)][2] =  1  x(N-1)

                                                          dx           N

 

                


[1] The dx notation was created by German mathematician Gottfried Wihlelm Leibniz.

[2] Remember, that in the equation y=f(x), we are differentiating the function y with respect to its independent variable, x.