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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 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 any point A on the line to any point B, the slope is the number

Δy
Δx

Change in y-coördinate
Change in x-coördinate

(Topic 8 of Precalculus.)

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 ?" If we can name the slope of a tangent line to the curve at that point, then that will be the answer. 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 one for finding what is called the derivative.

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. Then the slope of the secant is the average rate of change between those two points. For example, if

then on changing from x₁ to x₂, the 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 exactly at x₁? What is the slope of

the tangent to the curve at P?

We cannot, of course, evaluate

Δy
Δx

exactly at P -- because Δy

and Δx

would then both be 0, and the value of

Δy
Δx

would be completely

ambiguous.

Therefore we will consider shorter and shorter intervals of Δx, which will result in a sequence of secants --

-- a sequence of slopes. And we will define the tangent at P to be the limit of that sequence

That slope, that limit, 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 coördinates 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 new x-coördinate is x + Δx.
It is the x-coördinate of Q on the graph.

But when the value of x changes, there is a corresponding change Δy
in the value of y, that is, in the value of f(x). Its new value is f(x + Δx). The coördinates of Q are (x + Δx , f(x + Δx)).

Then

Finally:

The slope of the tangent line at P
is the limit of the ratio of the change in the function
to the change in the independent variable,
as that change 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."

Since the theorems on limits make it easy to evaluate a limit, it is easy to lose sight of its actual meaning. With regard to the derivative, it is the number which that ratio can approach as closely as we desire.

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, 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 f '(x) -- "f-prime of x" -- and we say that f itself 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, and we are to regard x as being fixed. It is the specific value at which we are evaluating 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 simplify:

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 definition 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 have to algebraically manipulate the difference quotient so that we can simply replace h with 0. In fact, the entire theory of limits, with all its complexities and subtleties, was invented to justify just that (Poor Newton and Leibniz were criticized for offering justifications that the inventors of limits didn't like.) We may put h = 0 here, because the difference quotient reduces to 2x + h, and is therefore a polynomial in 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.

(Conversely, 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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Think about this yourself first!

Notations for the derivative

Since the derivative is this limit: