Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 2: The Derivative – Section 2.3: The Derivative

 

2.3
The Derivative

 

 

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1. Rates Of Change And Slopes

 

In Section 2.1, we see that the

slope of secant S and the slope of tangent T to the graph of f at x = a are:

 

 

respectively. In Section 2.2, we see that the average rate of change of f over [a, a + h] and the instantaneous rate of change of f at x = a are:

 

 

respectively, which are the same as the 2 quantities above.

 

So the geometric interpretation of the average rate of change of f over [a, a + h] is the slope of the secant thru points
(a, f(a)) and (a + h, f(a + h)), and the geometric interpretation of the instantaneous rate of change of f at x = a is the
slope of the tangent to the graph of
f at x = a.

 

Fig. 1.1

 

–  Instantaneous rate of change and slope of tangent 
    at the same point are the same quantity.

 

–  Derivative is instantaneous rate of change and slope
    of tangent line.

 

Using Two-Sided limits

 

Since the instantaneous rate of change is unique if it exists, the limit involved in its definition is a two-sided limit, not
one-sided limits, which may be different when both exist. Similarly, since the tangent line is unique if it exists, the limit
involved in the definition of its slope is a two-sided limit. So the
h in their definitions must take on both positive and
negative values. In the above discussion, we implicitly assume
h > 0, so that a < a + h, the interval is [a, a + h], and the
limit is that of the difference quotient:

 

 

is valid for both positive and negative values of h.

 

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2. The Derivative

 

The slope of the tangent line to a straight line is the same at every point. The slope of the tangent line to the portion of
the graph of
y = f(x) displayed in Fig. 1.1 changes from point to point. We've got a quantity, the slope of the tangent line,
that may change, ie, we've got a function. Thus, it's useful to give it a technical name. That name is derivative, which
“
derives from the verb derive ”.

 

Definitions 2.1

 

 

This limit is called the derivative of f or the derivative of f with respect to x at a. (If it exists, then the derivative exists; if it doesn't exist, then the derivative doesn't exist.) See Fig. 1.1. The derivative f '(x) of f at any point x in the
domain of
f is defined in the same way, ie:

 

 

The quantity f '(x) may depend on x, so that f '(x) is a function of x. The function f '(x) is called the derivative of f
or the derivative of
f with respect to x. Note that the phrase "with respect to" precedes the variable with respect
to which the derivative is calculated.

 

If f '(a) exists, then f is said to be differentiable at a. If f is differentiable at every point in its domain, then it's
called a differentiable function.

 

To find the derivative f ' of a function f is called to differentiate f. The process of differentiating a function is called
differentiation.

 

 

Remarks 2.1

 

 

ii.  The derivative f '(a) is the instantaneous rate of change of f at x = a and the slope of the tangent line to the graph of f
   
 at x = a.

 

iii.  The derivative f '(a) is the value of the derivative (function) f '(x) at x = a.

 

iv.  The derivative of a function f is a function that derives from f, hence the name derivative.

 

 

 

3. Derivatives Of Some Elementary Functions

 

We now prove that the derivatives of the functions in the first column are as shown in the second column in the following
table, where
sgn, read “sig-nem”, is the signum (Latin for “sign”) function and is defined by:

 

 

 

 

 

Proof
For f(x) = C:

 

 

 

For f(x) = |x|. If x > 0 then f(x) = x so that f '(x) = 1 ( f(x) = ax + b, a = 1 and b = 0). If x < 0 then f(x) = –x so
that
f '(x) = –1. Hence f '(x) = sgn x.
EOP

 

Note that from Section 2.1.2 “Where There Are No Tangent Lines” it follows that the function f(x) = |x| isn't
differentiable at
x = 0. So we leave sgn x undefined at x = 0.

 

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4. Notations For Derivatives

 

The derivative of the function y = f(x) is denoted in several ways as follows:

 

 

The Leibniz Notation

 

The notation:

 

 

is called the Leibniz notation. It's read “ derivative of y with respect to x ”, or, for short, “ dee y over dee x ”. It's
suggestive of the fact that the derivative of
y with respect to x is the rate of change of y with respect to x. For now,
regard it as a  single symbol, not as a quotient. The quotient interpretation will appear in later sections and chapters.

 

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5. Equations Of Tangent Lines

 

Recall that the equation of a straight line passing thru the point (x0, y0) and having slope m is y = m(x – x0) + y0; see
Section 2.1.2. Hence the equation of the tangent line to the graph of
f at x = x0 is:

 

 y = f '(x0)(x – x0) + f(x0).

 

The equation of the tangent line T in Fig. 1.1 is y = f '(a)(x – a) + f(a).

 

Example 5.1

 

Find the equation of the tangent line to the curve y = x2 at x = 3.

 

Solution
The slope of the tangent line is y'|x=3 = 2x|x=3 = 2(3) = 6. When x = 3 we have y = 32 = 9. The equation of the tangent
line is
y = 6(x – 3) + 9, or y = 6x – 9.
EOS

 

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6. Obtaining Graphs Of Derivatives From Those Of Functions

 

Let f be differentiable on (a, b). See Fig. 6.1. The graph of f opens upward. We note that as x increases from a to b, the
tangent line to the graph of f at x turns counterclockwise, which means that its slope and thus the derivative f '(x)
 

Fig. 6.1

 

The graph of f opens upwayd on (a, b).

 

Fig. 6.2

 

The graph of g opens downward on (a, b).

 

increase. Let g be differentiable on (a, b). See Fig. 6.2. The graph of g opens downward. We note that as x increases
from a to b, the tangent line to the graph of g at x turns clockwise, which means that its slope and thus the derivative
g'(x) decrease.

 

Suppose we wish to sketch a graph of the derivative f ' of a function f on an interval [a, b] by using the graph of f on [a,
b]. See Fig. 6.3. For this task, we find an enough number of points (x, f '(x)) and join them by a curve that will be the
 

Fig. 6.3

 

Graph Of y = f(x).

 

Fig. 6.4

 

Obtaining Graph Of y = f '(x) From
That Of
y = f(x).

 

graph of f '. Since at any point x of [a, b] the value f '(x) is the slope of the tangent line to the graph of f at x, we select
a number of points (
x, f(x)) on the graph of f and at each of them estimate the slope f '(x) of the tangent line to the
graph of
f at that point, then we plot the point (x, f '(x)). Refer to Fig. 6.4. The selected points include:

 

–  the endpoints of the interval,
–  the
x-intercepts and y-intercept of the graph of f,
–  the points (
x, f(x)) where the tangent line to the graph of f is horizontal and thus has slope f '(x) = 0 so that the point
    (
x, 0) is the x-intercept of the graph of f ', and
–  points in each sub-interval formed by these
x-intercepts of the graph of f ' where the graph of f changes from opening
    upward to opening downward or respectively from opening downward to opening upward so that
f '(x) changes from
    increasing to decreasing or respectively from decreasing to increasing there and hence the graph of
f ' has a maximum
    or respectively a minimum there; at each of such points the tangent line to the graph of
f crosses the graph of f.

 

Of course more points on the graph of f have to be selected if it's necessary to do so for the shape of the graph of f ' to
be revealed.

 

To estimate the slope of the tangent line to the graph of f at a point x, we draw a short tangent line to the graph of f at
the point (
x, f(x)), draw a short horizontal line segment from the point (x, f(x)) to the right with length 1, and draw a
vertical line segment from the right endpoint of that horizontal line segment to the tangent line, then, as slope = rise/run
= rise/1 = rise, we estimate the signed length of that vertical line segment, which is positive if the vertical line segment is
upward toward the tangent line or negative if it's downward, and that's the desired slope.

 

After getting enough points (x, f '(x)), we join them together by a curve, which is the graph of f '.

 

Let y = f(x). Suppose x is increasing on a sub-interval, so that the change in x is positive on that sub-interval. If y is also
increasing, then the change in
y is also positive, therefore (change in y)/(change in x) > 0, and it must be that f '(x) =
dy/dx > 0 at any point x of the sub-interval, thinking of dx as a small change in x and dy as the corresponding small
change in
y. If y is instead decreasing, then the change in y is negative, therefore (change in y)/(change in x) < 0, and it
must be that
f '(x) = dy/dx < 0 at any point x of the sub-interval. If y is constant, then the change in y is 0, therefore
(change in
y)/(change in x) = 0, and it must be that f '(x) = dy/dx = 0 at any point x of the sub-interval. Thus make
sure that the following verification is satisfied:

 

–  on any sub-interval where f is increasing, the graph of f ' is above the x-axis,
–  on any sub-interval where
f is decreasing, the graph of f ' is below the x-axis, and
–  on any sub-interval where
f is constant, the graph of f ' is a line segment on the x-axis.

Example 6.1

 

A graph of y = f(x) is given in Fig. 6.5. Use it to sketch a graph of the derivative y = f '(x).

 

Fig. 6.5

 

Graph Of y = f(x).

 

Solution
The graph of y = f '(x) as obtained from that of y = f(x) is sketched in Fig. 6.6.

 

Fig. 6.6

 

Graph Of y = f '(x) Obtained From
That Of
y = f(x).

EOS

 

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Problems & Solutions

 

1.  i.  Using a calculator, find the slope of the secant line to f(x) = x3 – 2x passing thru the points corresponding to x = 2
         and
x = 2 + h, where:

 

         a.   h = 0.1.
         b.
   h = 0.01.
         c.  
h = 0.001.

 

    ii.   Guess f '(2).
    iii.  Calculate
f '(2) directly from the definition of derivative.

 

Solution

 

 

Note that f(2) = 23 – 2(2) = 4.

 

 

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2.  Differentiate each of the following functions directly from the definition of the derivative.

 

     a.  f(x) = 1 + 4x – 5x2.
     b. 
g(t) = (t2 – 3)/(t2 + 3).
     c. 
h(x) = x + |x – 1|.

 

Solution

 

 

c.  The graph of y = h(x) = x + |x – 1| has a sharp point at x = 1. We'll show that h has a derivative everywhere on R
     except at
x = 1.

 

    

 

     If x – 1 > 0 or x > 1, then | x – 1| = x – 1, so h(x) = x + (x – 1) = 2x – 1, thus:

 

    

 

    

 

     h'(1) doesn't exist.

 

Notes

 

 

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3.  Calculate:

 

   

 

    directly from the definition of the derivative.

 

Solution

 

Let f(x) = x1/3. Then:

 

 

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4.  Utilizing the definition of derivative, find an equation of the tangent line to the curve y = 2/(t2 + t) at the point where
    
t = 1.

 

Solution

 

At t = 1 we have y = 1. The slope of the tangent line at t = 1 is:

 

 

Note

 

The notation y(E ), where E is an expression, means the value of y at t = E, not the product yE. This notation is similar
to the notation
f(t). It treats y as a function of t. Similarly for the notation y'(E ).

 

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5.  The graph of f is given in the figure below. Use it to sketch a graph of the derivative f '.

 

    

 

Solution

 

 

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6.  Prove that:

 

     a.  The derivative of an even differentiable function is odd.
   
 b.  The derivative of an odd differentiable function is even.

 

Solution

 

a.  Let f be an even differentiable function and x an arbitrary point in its domain. Then:

 

 

     Thus, g' is even.

 

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