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INCREMENTS AND DIFFERENTIATION In this section we will extend our discussion of limits
and examine the idea of the derivative, the basis of differential calculus. We
will assume we have a particular function of x, such that _{ } If x is assigned the value 10, the corresponding value of
y will be (10)^{2} or 100. Now, if we increase the value of x by 2,
making it 12, we may call this increase of 2 an increment or
x. This
results in an increase in the value of y, and we may call this increase an
increment or
y. From
this we write
As x increases from 10 to 12, y increases from 100 to 144
so that
and
We are interested in the ratio
because the
limit of this ratio as
x
approaches zero is the derivative of y = f(X) As you recall from the discussion of limits, as
x is
made smaller,
y gets
smaller also. For our problem, the ratio
approaches 20. This is shown in
table 41. Table 4l.Slope Values
We may use a much simpler way to find that the limit of
as
x approaches zero is, in this case, equal to 20. We
have two equations
and
By expanding the first equation so that
and subtracting the second from this, we have
Dividing both sides of the equation by
x gives
Now, taking the limit as
x
approaches zero, gives
Thus,
NOTE: Equation (1) is one way of expressing the derivative
of y with respect to x. Other ways are
Equation (1) has the advantage that it is exact and true
for all values of x. Thus if x=10 then
and if x=3 then
This
method for obtaining the derivative of y with respect to x is general and may
be formulated as follows: 1.
Set up the function of x as a function of (x +
x)
and expand this function. 2.
Subtract the original function of x from the new function of (x +
x). 3.
Divide both sides of the equation by
x. 4.Take the limit of all the terms in the equation as x approaches zero. The resulting equation is the derivative of f(x) with respect to x. 
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