Theme: First Derivative and First Differential

Theme: First Derivative and First Differential

Plan: A derivative. One-sided derivatives. The rules for calculating derivatives. A differential. The rules for calculating differentials. The formulae for differentiating some elementary functions. A normal to the curve. Invariance of the form of the first differential.

The finite limit of the ratio of the function increment to the increment of the argument, when the latter tends to zero, is called a derivative of the function with respect to the argument x:

, or .

A differential of a function is the linear (respectively ) part of the increment of the function. It is denoted as or and .

Finding a derivative is called differentiating a function.

One-sided derivatives

If () exists, then it is called a right (left) derivative of the function at point and denoted as (respectively ).

If and exist and they are equal, then exists, and it is equal . Visa versa, if exists, then and exist, and = = .

The rules for calculating derivatives

Derivative of a Sum [The Sum Rule]:

(f (x) ± g (x))′ = f′ (x) ± g′ (x).

Derivative of a Product [The Product Rule]:

[ f (x) · g (x)]′ = f′ (x) · g (x) + f (x) · g′ (x).

Derivative of a Quotient [The Quotient Rule]:

Derivative of a Composition [The Chain Rule]:

[ f ο g (x)]′ = f′ (g (x)) · g′ (x).

The formulae for differentiating some elementary functions

A normal to the curve is the straight line, which is perpendicular to the tangent line and which cuts the point of touching.

The equation of the normal has the form .

An angle between two curves and in their intersection point is the angle between the tangent lines for these curves at the point .

.

If there defined a law of the straight-lined movement , then the velocity of the movement at the moment is a derivative of the distance with respect to time: .

Invariance of the form of the first differential

Suppose the argument x of the differentiable at point function is not an independent variable, but a function of some independent variable t: , and , and is differentiable at point . Then the differential of the function has the form: , but now is not an arbitrary increment of the argument x (as it was in the case, when x is an independent variable), but a differential of the function at point , i.e. . This property is called an invariance of the form of the first differential.

Дата добавления: 2015-11-04