One of the
most important concepts of calculus is the derivative. The derivative of a
function f(x), represents the tangent line in a specified point of the curve
f(x). Many problems and phenomena of the nature can be solved by finding the
tangent line to a curve that describes the behavior of such phenomena.
In physics
there are many applications for derivative, as rectilinear motion, oscillatory motion
and electrostatic applications.
If you want
to understand the concept, first of all you have to know about the theories of
limits, because derivative is a limit, properly said, limits are an elementary
part of calculus, so I think it won’t be necessary to exhibit here; so I won’t
develop so much than applying it in application models because it is so
elementary that I will assume you know about; and also I think it is bit
boring; however you have to know limits are the basis of calculus.
So well
first of all, I want to introduce the basic concept of derivative, but before
we have to remember the basis concept of a tangent line. Surely you will know
what a tangent to a circle is, and you will know is easy to establish it; so
well we’re going to take the concept to a most general application, we’re going
to try to find the tangent line for either cure.
Take a look
to the next figure.
At the
figure we have a curve in red that represents a function that depends of the
variable ‘’x’’. So if we choose two points ‘”P” and “Q” located on the curve,
we can join those points by using a line, (presented in blue on the figure),
and that line would be a secant line for the cure in the figure.
So the
slope of this secant line can be expressed as follows:
m = f(x2)-f(x1)
x2-x1
And we can
set x2-x1 = h, and due to this, the slope can be also expressed
as:
m = f(x1+h)-f(x1)
h
Now let “P”
without moving and let “Q” moving along the curve getting closer to “P”. And as
we can see at the next figure the interval h is reducing.
The new
secant to the curve now in green, also we can show its initial position in blue;
so if we continue reducing the interval
h until it approaches to zero, the secant line becomes a tangent for the curve,
and this occurs in the limit, and mathematically it can be expressed as
follows:
f'(x) = lim f(x1+h)-f(x1)
h->0 h
Finally
we have introduced the essential definition of derivative. From here all rules
and theorems comes out. As I said you before the derivative is an undefined
limit, that should be solved by using algebraic procedures, but also can be
used L’H^opital rule, is easier but involves the derivative, and using
derivative to find derivative is redundant.
However
there are many derivation formulas, those formulas comes from the general one
seen above and for practical purposes we’re going to uses these new formulas;
the most general one (the power rule for derivation) is presented in the next
link with its deduction from the general formula, so you are able to watch it
or downloaded.
Power Rule: http://www.4shared.com/photo/A6Clzs7S/regla_de_la_potencia_1.html
So if we
have for example f(x) = x1/2 you can find the derivative by multiplying the
power of the variable by the coefficient, and then subtracting 1 from the
power, and so you will get: 1/x1/2, in the next link I left the deduction of
this specified case by using the general formula for you to compare:
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