5
THE DERIVATIVE
The rate of change of a function
at a specific value of x
The slope of a tangent line to a curve
The definition of the derivative
The derivative of f(x) = 2x − 5
The equation of a tangent to a curve
Now the slope of a straight line indicates a constant rate of change as
we move from any point A on the line to any point B. (Topic 8 of Precalculus.) The slope is the number
| Δy Δx |
= |  |
= | Change in y-coordinate Change in x-coordinate |
. |
That number indicates how the value of y changes when the value of x changes. Δy/Δx is constant. A straight line has one and only one slope.
If x represents time, for example, and y represents distance, then a
straight line graph that relates them indicates constant speed. 45 miles per hour, say -- at every moment of time.
The slope of a tangent line to a curve
Calculus however is concerned with rates of change that are not constant.
If this curve represents distance Y versus time X, then the rate of change — the speed — at each moment of time is not constant. The question that calculus asks is: "What is the rate of change at exactly the point P ?" The answer will be the slope of a tangent line to the curve at precisely that point. And the method for finding that slope — that number — was the remarkable discovery by both Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716). That method is the fundamental procedure of differential calculus.
A secant to a curve
A secant is a straight line that cuts a curve. (A tangent is a straight line that just touches a curve.) Hence, consider a secant line that cuts the curve at points P and Q. The slope of the secant is the average rate of change between those two points. For example, if
then on going from x1 to x2, that function has changed an average of 4 units of y for every 5 units of x. But once again, the question calculus asks is: How is the function changing precisely at x1?
What is the slope of the tangent to the curve precisely at P?
| To answer, we must try to evaluate the slope, |  |
, as Q approaches P |
along the curve, to the point of coinciding with it.
The sequence of secants becomes the tangent at P 
Arithmetically, we must evaluate the slope of the secant as Δx becomes closer and closer to 0. But as Δx approaches 0, so does Δy. We will be trying to evaluate their ratio as both of them become vanishingly small. How can we possibly do that? Only by considering the limit of a sequence of their ratios. That is,
That slope will be the value of what we will call the "derivative."
The definition of the derivative
Let y = f(x) be a continuous function, and let the coordinates of a fixed point P on the graph be (x, f(x)). (Topic 4 of Precalculus.) Let x now change by an amount Δx. Then the x-coordinate of the change is
x + Δx. But when the value of x changes, there is a corresponding change Δy in the value of f(x). Its new value is f(x + Δx). The coordinates of Q are (x + Δx , f(x + Δx)).
Then
Therefore the slope of the tangent line at P will be the limit as Δx approaches 0.
Since Δx -- not x -- is the variable that approaches 0, x will remain constant, and that limit will be a function of x. Since it will be derived from f(x), we call it the derived function or the derivative of f(x) And to remind us that it was derived from f(x), we denote it by f '(x) -- "f-prime of x."
This quotient --
-- is called the Newton quotient, or the difference quotient. Calculating and simplifying it is a fundamental task in differential calculus.
Again, the difference quotient is a function of Δx. But to simplify our written calculations, then instead of writing Δx, we will write h.
| Δx | = | h |
| Δy | = | f (x + h) − f (x) |
The difference quotient then becomes
We now express the definition of the derivative as follows.
DEFINITION 5. By the derivative of a function f(x), we mean the following limit, if it exists:
We call that limit the function f '(x) -- "f-prime of x" -- and we say that f is differentiable at x, and that f has a derivative.
Again, in taking that limit, the variable that is approaching 0 is h, not x. Because a limit is by definition a number, we must regard x as being fixed. It is the specific value at which we are evalulating the rate of change of f(x).
In practice, we have to simplify the difference quotient before letting h approach 0. We have to express the numerator --
f (x + h) − f (x)
-- in such a way that we can divide it by h.
As an example, we will apply the definition to prove the following:
| THEOREM. | f(x) | = | x² |
| implies | |||
| f '(x) | = | 2x. | |
Proof. Here is the difference quotient, which we will proceed to simplfy:
| 1) | (x + h)² − x² h |
|
| 2) | = | x² + 2xh + h² − x² h |
| 3) | = | 2xh + h² h |
| 4) | = | 2x + h. |
In going from line 1) to line 2), we squared the binomial x + h. (Lesson 18 of Algebra.)
In going to line 3), we subtracted the x²s. That is, we subtracted f(x).
In going to line 4), we divided the numerator by h. (Lesson 20 of Algebra.)
We can do that because h is never equal to 0, even when we take the limit (Lesson 2).
We now complete the defintion of the derivative and take the limit:
| f '(x) | = |  |
(2x + h) |
| = | 2x. | ||
This is what we wanted to prove.
Whenever we apply the definition, we must express the difference quotient in such a way that we can evaluate the limit simply by replacing h with 0 For, the difference quotient -- in this case, 2x + h -- will be a continuous function of h.
Differentiable at x
According to the definition, a function will be differentiable at x if a certain limit exists there. Graphically, this means that the graph at that value of x will have a tangent line. At which values, then, would a function not be differentiable?
Where it does not have a tangent line
Above are two examples. The function on the left does not have a derivative at x = 0, because the function is discontinuous there. At x = 0 there is obviously no tangent.
As for the graph on the right, it is the absolute value function, y = |x|. (Topic 5 of Precalculus.) And it is not possible to define the tangent line at x = 0, because the graph makes an acute angle there. In fact, the slope of the tangent line as x approaches 0 from the left, is −1. The slope approaching from the right, however, is +1. The slope of the tangent line at 0 -- which would be the derivative at x = 0 -- therefore does not exist . (Definition 2.2.)
The absolute value function nevertheless is continuous at x = 0. For, the left-hand limit of the function itself as x approaches 0 is equal to the right-hand limit, namely 0. This illustrates that continuity at a point is no guarantee of differentiability -- the existence of a tangent -- at that point.
(Conversly, though, if a function is differentiable at a point -- if there is a tangent -- it will also be continuous there. The graph will be smooth and have no break.)
Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all values of their domains. Such functions are called differentiable functions.
Can you name an elementary class of differentiable functions?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Think about this yourself first!
Polynomials.
Notations for the derivative
| Since the derivative is this limit: |  |
then a symbol for the |
| derivative is |  |
(Read: "dee-y, dee-x.") |
For example, if
| y | = | x², | |
| then, as we have seen, | |||
 |
= | 2x. | |
"Dee-y, dee-x (the derivative of y with respect to x) is 2x."
We also write
y '(x) = 2x.
"y-prime of x is equal to 2x."
| This symbol by itself: | d dx |
("dee, dee-x") | , is called the differentiating |
operator. We are to take the derivative of what follows it. For example,
| d dx |
f(x) indicates the derivative with respect to x of f(x). |
| d dx |
(4x3 − 5) indicates the derivative with respect to x of (4x3 − 5). |
And so on.
A simple difference quotient
| The difference quotient is a version of | Δy Δx |
. And at times we |
will use that simple version. That is, the change in the value of a function y = f(x) is y + Δy. Hence the difference quotient is
At times it will be convenient to express the difference quotient as
Note: As Δx approaches 0 -- as the point Q moves closer to P along the curve -- then Δy, or equivalently, Δf also approaches 0. That is,
The student should now do Problems that require the definition of the derivative.
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