The derivative of a function of _x_ is the limit towards which tends the ratio of the increment of the function to the increment _h_ of the variable, in proportion as _h_ tends towards zero. Derivatives of trigonometric functions. Derivatives of exponentials and of logarithms. Rules to find the derivative of a sum, of a product, of a power, of a quotient of functions of _x_, the derivatives of which are known. _Of the numerical resolution of equations._ Changes experienced by an entire function _f(x)_ when _x_ varies in a continuous manner.--When two numbers _a_ and _b_ substituted in an entire function _f(x)_ give results with contrary signs, the equation _f(x) = 0_ has at least one real root not comprised between _a_ and _b_. This property subsists for every species of function which remains continuous for all the values of _x_ comprised between _a_ and _b_. An algebraic equation of uneven degree has at least one real root.--An algebraic equation of even degree, whose last term is negative, has at least two real roots. Every equation _f(x) = 0_, with coefficients either real or imaginary of the form _a + b √-1_, admits of a real or imaginary root of the same form. [Only the enunciation, and not the demonstration of this theorem, is required.] If _a_ is a root of an algebraic equation, the first member is divisible by _x - a_. An algebraic equation of the _m_^{th} degree has always _m_ roots real or imaginary, and it cannot admit more.--Decomposition of the first members into factors of the first degree. Relations between the coefficients of an algebraic equation and its roots. When an algebraic equation whose coefficients are real, admits an imaginary root of the form _a + b √-1_, it has also for a root the conjugate expression _a - b √-1_. In an algebraic expression, complete or incomplete, the number of the positive roots cannot surpass the number of the variations; consequence, for negative roots. Investigation of the product of the factors of the first degree common to two entire functions of _x_.--Determination of the roots common to two equations, the first members of which are entire functions of the unknown quantity. By what character to recognize that an algebraic equation has equal roots.--How we then bring its resolution to that of several others of lower degree and of unequal roots. * * * Investigation of the commensurable roots of an algebraic equation with entire coefficients. When a series of equidistant numbers is substituted in an entire function of the _m_^{th} degree, and differences of different orders between the results are formed, the differences of the _m_^{th} order are constant.