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GENERAL FORMULA To obtain a formula for the derivative of any expression
in x, assume the function
so that
Subtracting equation (4.2) from equation (4.3) gives
and
dividing both sides of the equation by
x,
we have
The
desired formula is obtained by taking the limit of both sides as
x
approaches zero so that
or
NOTE:
The notation
is not to be considered as a fraction in
which dy is the numerator and dx is the denominator. The expression
is
a fraction with
y
as its numerator and
x
as its denominator. Whereas,
is a symbol representing the limit approached
by
as
x
approaches zero. EXAMPLES
OF DIFFERENTIATION In
this last section of the chapter, we will use several examples of
differentiation to obtain a firm understanding of the general formula. EXAMPLE:
Find the derivative,
^{},^{ }for the function
determine
the slopes of the tangent lines to the curve at
and
draw the graph of the function. SOLUTION: Finding the derivative by formula, we have
and
Expand equation (1), then subtract equation (2) from
equation (1) , and simplify to obtain
Dividing both sides by Ax, we have
Take the limit of both sides as
Then
The slopes of the tangent lines to the curve at the points
given, using this derivative, are shown in figure 47, view B. Thus we have a new method of graphing an equation. By
substituting different values of x in equation (3), we can find the slope of
the tangent line to the curve at the point corresponding to the value of x. The
graph of the curve is shown in figure 47, view A. EXAMPLE: Differentiate the function; that is, find
of
and then find the slope of the tangent line to the curve
at SOLUTION:
Apply the formula for the derivative, and simplify
as follows:
Now take the limit of both sides as
so that
To find the slope of the tangent line to the curve at the
point where x has the value 2, substitute 2 for x in the
expression for
.
EXAMPLE: Find the slope of the tangent line to the curve
At x=3 SOLUTION:
We need to find
, which is
the slope of the tangent line at a given point. Apply the formula for the
derivative as follows:
and
Expand equation (1) so that
Then subtract equation (2) from equation (1): _{
} Now, divide both sides by
^{
} Then take the limit of both sides as
Substitute 3 for x in the expression for the derivative to
find the slope of the tangent line at x=3 so that slope = 6 In this last example we will set the derivative of the
function, f (x), equal to zero and determine the values of the independent
variable that will make the derivative equal to zero to find a maximum or minimum
point on the curve. By maximum or minimum of a curve, we mean the point or
points through which the slope of the tangent line to the curve changes from
positive to negative or from negative to positive. NOTE:
When the derivative of a function is set
equal to zero, that does not mean in all cases we will have found a maximum or
minimum point on the curve. A complete discussion of maxima or minima may
be found in most calculus texts. To
set the derivative equal to zero, we will require that the following conditions
be met: 1.
We have a maximum or minimum point. 2.
The derivative exists. 3.
We are dealing with an interior point on the curve. When
these conditions are met, the derivative of the function will be equal to zero. EXAMPLE: Find the derivative of the function
set
the derivative equal to zero, and find the points of maximum and minimum on the
curve. Then verify this by drawing the graph of the curve. SOLUTION: Apply the formula for
as follows:
and
Expand
equation (1) and subtract equation (2), obtaining
Now,
divide both sides by
, and
take the limit as
so that
Set
equal to
zero; thus
Then
and
Set each factor equal to zero and find the points of
maximum or minimum; that is,
and x=1 The graph of the function is shown in figure 48.
