ISAAC NEWTON AND GOTTFRIED LEIBNIZ

IISAAC NEWTON (1642 - 1727)

British mathematician and scientific, and co-inventor of calculus, in a two years period (1642 – 1727), Newton made important  and fundamental contribution to calculus and optical, besides being the discovered of the universal gravitation law,  he did not publish his discoveries from the first moment, only introduced his results as  useful tools into his scientific papers.

His work “mathematical principles of natural philosophy” is considered as one of the greatest achievements of human thought.

Gottfried Leibniz  (1646 - 1716)

German mathematician and philosopher who introduced much of the terminology and notation of calculus, terminology that was created by the same, at the same time that Sir Isaac Newton. At 20 years old, Leibniz was a prodigal, he was a lawyer and had published papers about logic and jurisprudence, He was a real man of Renaissance, and he was possessor of an extraordinary library, and contributed to politics, to philosophy, technology, engineering, linguistics, geology, architecture and physics. In mathematics, he deduced fundamental rules for derivation and promoted the development of calculus through its extensive communications by letter. The notation he presented are logical and simple, and have been barely improved in the last 300 years, these notation make calculus more accessible. 

One of the phrases he wrote:

"The symbols shown an advantage in discoveries, even greater when they express with brevity the exact nature of something . . . , thus decreasing the effort of thought." 

FUNDAMENTALS OF DIFFERENTIAL CALCULUS OF REAL VARIABLES

 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:

http://www.4shared.com/photo/VMrXdWuh/ejemplo_2.html