The meaning of the derivative - An approach to calculus

An Approach

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C A L C U L U S

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THE DERIVATIVE

The rate of change of a function
at a specific value of x

The slope of a straight line

The slope of a tangent line to a curve

A secant to a curve

The slope at a specific point

The definition of the derivative

The derivative of f(x)= x²

Differentiable at x

Notations for the derivative

A simple difference quotient

Section 2: Problems

The derivative of f(x) = 2x − 5

The equation of a tangent to a curve

The derivative of f(x) = x³

CALCULUS IS CONCERNED WITH THINGS that do not change at a constant rate. The values of the function called the derivative
will be that varying rate of change.

Now the slope of a straight line indicates a constant rate of change as

we move from a point A on the line to a point B; so many units of y for each unit of x. (Topic 8 of Precalculus.) The slope is the number

Δy
Δx

Change in y-coordinate
Change in x-coordinate

This rate of change 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 will ask is: How is the function changing precisely at x₁?

To answer, let the point Q approach the point P along the curve, and let us calculate the average rate of change as x₂ gets closer and closer and closer to x₁. That is, let x₂ approach x₁ as a limit (Lesson 2). In other words, let Δx approach 0.

We will then have a sequence of secant lines each with its own slope. And the limit of that sequence will be the slope m of the tangent line at P

The secant will revolve about the fixed point P, and the limit of its revolving is the tangent line. The slope of the tangent at that point will be the value of what we will call the "derivative".

The slope at a specific point

Let y = f(x), and let P be a fixed point on the graph whose coordinates are (x, f (x)). (Topic 4 of Precalculus.) Let Q be another point whose coordinates are (z , f (z)).

Then the slope of the secant line PQ is

Let Δx approach 0 as a limit, so that Q approaches P along the curve -- that is, let z approach x. Therefore the slope of the tangent line at P is

Now, to evaluate that limit for a specific function, we would have to divide the numerator by z − x. Let us do that for f(x) = x².

We first factored the difference of two squares (Lesson 19 of Algebra).

We then divided the numerator by z − x. (We may do that because z is never equal to x; that is, the denominator is never 0, even when we take the limit; Lesson 2).

And so we found the slope of the secant PQ to be z + x . To find the slope of the tangent at P, we will take the limit as z approaches x.

Now, z is the variable here, and z + x is a polynomial in z (Lesson 2). Therefore we may evaluate that limit simply by replacing z with x.

The slope, then, of the tangent to the graph of y = x² at the value x -- the rate of change of the function at that value -- is 2x. At x = 4, for example, the slope is 2· 4 = 8. At x = −5, the slope is 2· −5 = −10. And so on.

The value of the slope of the tangent is itself a function of x. In this case it is 2x. 2x is the rate of change of f(x) = x² at each value of x.

Now, since 2x was derived from f(x) = x², 2x is called the derived function, or the derivative, of x². To remind us that it came from f(x), we write f '(x) -- "f-prime of x."

f(x)	=	x²

f '(x)	=	2x.

The definition of the derivative

That limit which is the slope of the tangent at (x, f (x)), is a definition of the derivative, f '(x):

However, it is not the standard definition. But it will become the standard definition by replacing z with x + Δx, where Δx ia a change in the value of x.

Again, let y = f(x), and let the coordinates of a fixed point P on the graph be (x, f(x)). The coordinates of Q are (x + Δx , f(x + Δx)).

Then

Therefore the slope of the tangent line at P -- the function we are calling f ' (x), the derivative of f(x) -- is the following limit as Q approaches P along the curve:

This quotient --

-- is called the Newton quotient, or the difference quotient. Calculating and simplifying it is a fundamental task in differential calculus.

Now the difference quotient is a function of Δx. To simplify our 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 5.1. 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.

Note: In taking that limit, the variable is h. Upon evaluating the limit, it then becomes a function of x, which we must think of as being fixed. It is the specific value of x at which we can evalulate 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, that 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?

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Notations for the derivative

Since the derivative is this limit: