5

THE DERIVATIVE

The rate of change of a function
at a specific point

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 a point

Notations for the derivative

A simple difference quotient

Section 2:  Problems

The derivative of f(x) = 2x − 5

The derivative of f(x) = x³

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

To answer, let the point Q approach the point P, 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 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

Let y = f(x).  Let x be a specific point in the domain of f, and let z be another point.  Then the coordinates of P are (x,  f (x)), and the coordinates of Q are (z ,  f (z)).  (Topic 4 of Precalculus.)

Therefore, the slope of the secant line PQ is

Let Δx approach 0 -- that is, let z approach x.  Then the slope of the tangent line at P is

To be explicit, let us evaluate that limit for the function f(x) = x².

Then the coordinates of P are (x, x²), and the coordinates of Q are (z , z²).

First, we will evaluate the slope of the secant line PQ:

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).

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.

But 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 y = x² at the point x -- the rate of change of the function at that point -- 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 is itself a function of x.  In this case it is 2x.  2x is the rate of change of  f(x) = x² at the specific point 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."

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:

The difference quotient then becomes

We will therefore express the definition of the derivative as follows.

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.

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.

According to the definition, a function will be differentiable at a point  if a certain limit exists there.  Graphically, this means that the graph at that point will have a tangent line.  At which points, then, would a function not be differentiable?

Where it does not have a tangent line

(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.)

The graph on the right is the absolute value function. (Topic 5 of Precalculus.)  Approaching 0 from the left, the slope of the tangent line -- which is the graph itself -- is −1.  The slope approaching from the right, however, is +1.  The limit at 0 -- which would be the derivative -- therefore does not exist . (Definition 2.2.)

Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all points 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

For example, if

y		=	x²,
then, as we have seen,
		=	2x.

"The derivative of y with respect to x is 2x."

We also write

y ' = 2x.

"y-prime is equal to 2x."

This symbol by itself:

d  dx

, 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.

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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