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4.3. Tangents
If a curve C has equation y = f(x) and we want to find the tangent
to C at the point P (x, f(x)), then we consider a nearby point Q(x+
∆x, f(x +∆x)) where ∆x =0and compute the slope of the secant
line PQ:
m
PQ
=
f(x +∆x) − f(x)
∆x
.
Then we let Q approach P along the curve C by letting ∆x approach
0.Iftheslopem
PQ
approaches a number m,thenwedefine the
tangent T to be the line through P with slope m. (This amounts to
saying that the tangent line is the limiting position of the secant line
PQ as Q approaches P ).
Remark 2
Geometrically, the slope equals tan ϕ,whereϕ is an angle between
the positive x-semiaxis and the tangent T .
4.4. Other notations
Some common alternative notations for the derivative are as follows:
f
(x)=y
=
dy
dx
=
d
dx
f(x)=D
x
f(x).
The symbols D
x
and
d
dx
are called differential operators because
they indicate the operation of differentiation, which is the process
of calculating a derivative. The symbol
dy
dx
which was introduced
by Leibniz, should not be regarded as a ratio (for the time being);
it is simply a synonym for f
(x). Nonetheless, it is a very useful
and suggestive notation, especially when used in conjunction with
increment notation.
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