Application of derivative in investigation of functions Continuity and differentiability of function. Sufficient conditions of functions monotony. Darboux's theorem. Intervals of function monotony. Critical points. Extreme ( minimum, maximum ). Points of extreme. Necessary condition of extreme. Sufficient conditions of extreme. Plan of function investigation. … [Read more...]

## De L’Hospital’s rule

De L'Hospital’s rule Let at x a for functions f ( x ) and g ( x ), differentiable in some neighborhood of the point a , the conditions are executed: This theorem is called de L'Hospital’s rule. It allows to calculate limits of ratios of functions, when both a numerator and a denominator approach either zero, or infinity. As mathematicians say, de … [Read more...]

## Derivatives of elementary functions

Derivatives of elementary functions ( x n ) ’ = n x n - 1 , ( n - a natural number ) ; ( sin x ) ’ = cos x ; ( cos x ) ’ = - sin x ; Back … [Read more...]

## Basic properties of derivatives and differentials

Basic properties of derivatives and differentials Properties of derivatives and differentials. Derivative of a composite function. If u ( x ) ≡ const , then u’ ( x ) ≡ 0 , du ≡ 0. If u ( x ) and v ( x ) are differentiable functions at a point x0 , then: ( c u )’ = c u’ , d ( c u ) = c du , ( c – const ); ( u ± v … [Read more...]

## Differential and its relation with derivative

Differential and its relation with derivative Differential of a function. Geometrical meaning of a differential. Differential of a function is a product of a derivative f ’( x0 ) and an increment of argument : df = f ’( x0 ) · . Geometrical meaning of a differential is clear from Fig.2 : here df = CD Back … [Read more...]

## Derivative. Geometrical and mechanical meaning of derivative

Derivative. Geometrical and mechanical meaning of derivativeDerivative. Argument and function increments. Differentiable function. Geometrical meaning of derivative. Slope of a tangent. Tangent equation.Mechanical meaning of derivative. Instantaneous velocity. Acceleration.Derivative. Consider a function y = f ( x ) at two points: x0 and x0 + : f ( x0 ) and f ( x0 + … [Read more...]

## Limits of functions

Limits of functions Limit of a function. Some remarkable limits. Infinitesimal and infinite values. Finite limit. Infinite limit. Notion of an infinity. Limit of a function. A number L is called a limit of a function y = f ( x ) as x tends a : if and only if for any > 0 one can find such a positive number = ( ), depending on , that | x – … [Read more...]

## Sequences. Limits of numerical sequences. Some remarkable limits

Sequences. Limits of numerical sequences. Some remarkable limits Numerical sequences. General term of a sequence. General term formula. Limit of numerical sequence. Convergent sequence. Divergent sequence. Bounded sequence. Monotone sequence. Weierstrass's theorem. Basic properties of limits. Some remarkable limits. Sequences. Consider the series of natural … [Read more...]