begin{equation*}

frac{d(ucdot v)}{dx} = ucdot frac{dv}{dx} + vcdot frac{du}{dx}

end{equation*}

Calculate the […]

]]>begin{eqnarray*}

text{Hyperbolic sine function }qquadsinh x &=& frac{e^{x} – e^{-x}}{2}

text{Hyperbolic cosine function }qquadcosh x &=& […]

begin{equation*}

intlimits_{a}^{b} f(x) cdot dx = F(b) – F(a)

end{equation*}

In its justification the telescopic sum works […]

]]>begin{equation*}

intlimits_{a}^{b} f(x)cdot dx

end{equation*}

represents the sum of an infinite quantity of infinitesimals, but no clue about how to […]

]]>Given the succession of numbers:

begin{equation*}

a_0, a_1, a_2, cdots, a_n

end{equation*}

Consider the succession of […]

]]>Motivation

The derivative of the function $y = F(x)$ was calculated and as a result the function $y = x^2$ was obtained.

What […]

Algebraic Functions

begin{eqnarray*}

displaystylefrac{dleft(u + vright)}{dx} &=& frac{du}{dx} + […]

Derivative of the function $y = arcsin x$

Let $y = arcsin x$. Then, $x = sin y$. […]

]]>In order to justify the formulas to calculate the derivative of the functions sine and cosine it is necessary to prove two preliminary results.

First the proof that $sin(dx) / dx = 1$ is […]

]]>begin{eqnarray*}

frac{f(x + dx) – f(x)}{dx} &=& frac{log_{a}(x + dx) – log_{a} (x)}{dx}

&=& frac{1}{dx} cdot […]

The following explanation to define the number $e$ is due Leonhard Euler (Source: Una relectura del Introductio in analysin infinitorum de Euler, by Quintero (1999).)

Consider the quantity […]

]]>We begin with the justification of the formula to calculate the derivative of a constant function

and then […] ]]>

begin{eqnarray*}

g(x + dx) &=& g(x) + g'(x)cdot dx = g(x) + dg quad […]

begin{eqnarray*}

frac{dy}{dx} = f'(x) &Rightarrow& dy = f'(x)cdot dx

&Rightarrow& f(x + dx) – f(x) = f'(x)cdot dx

&Rightarrow& f(x + dx) […]

begin{equation*}

m = frac{Delta […]

In this respect, Euler […]

]]>That is to say, an infinitesimal is an infinitely small quantity.

Infinitesimal

A quantity less than any […]

]]>begin{equation*}

A_{text{triangle}} = frac{1}{2},bcdot h

end{equation*}

it is obvious that two triangles having equal […]

]]>begin{tikzpicture}

coordinate (A) at (4,5);

coordinate (B) at (5,0);

coordinate (C) at (0,0);

coordinate (D) at (4,0);

%coodrinate (E) at ();

draw[thick] […]

begin{tikzpicture}

draw[thick] (0,0) — (3,0) — (5,4) — cycle;

draw[thick,dashed] (5,4) — (5,0) — (3,0);

node[above] at […]

begin{tikzpicture}

draw[thick] (0,0) — (5,0) node[below,midway]{$a$};

draw[thick] (0,0) — (4,5)node[above,midway]{$b$};

draw[thick] (4,5) — […]

Consider a sphere of radius $r$. […]

]]>