3. Higher Derivatives

Suppose that there is the differentiable function y=f(x), i.e., there is the derivative y=f(x), which is denoted by f₁(x) . By Definition of the derivative we obtain

.

Definition. The derivative of the derivative of function is called the second derivative; the derivative of the second derivative of function is called the derivative of the third order respectively and is denoted by

,

,

. . . . . . .

etc.

Notations:

, , etc.,

are read, for example, the third derivative – d three y with respect to the third power of dx.

If the order of derivative is noted in Arabic letters then it is written down in brackets.

In applications it is often required to find the higher derivatives of the product of two functions

(u^.v)⁽ⁿ⁾.

To derive this Leibniz formula, find the following derivatives of the first order

(u^.v)=uv+uv,

and of the second order

(u^.v)=uv+uv+uv+vu= uv+2uv+uv.

We differentiate once again and find the derivative of the third order:

(u^.v)=uv+uv+2uv+2uv+uv+uv= uv+3uv+3uv+uv.

The derivative of the derivative of two functions is said to be the Binomial theorem if the inverse order is taken nominally as the power of this function:

(u^.v)⁽ⁿ⁾=u^(n). v+C_n^1. u^(n-1). v+ C_n^2. u^(n-2). v+ . . . +

+ C_n^n-1.uv^(n-1)+C_n^n.uv⁽ⁿ⁾.

_{LECTURE 6}

Rolle's theorem, Lagrange's theorem, and Fermat's theorem. L'Hopital's rule. Investigation of the function. Extremum of the function. Convexity, concavity and point of inflection. Asymptotes

1. Properties of differentiable functions

Fermat's Theorem.

Let be defined and differentiable on an open interval (a, b). If attains its absolute maximum or absolute minimum (both are called absolute extremum) at , where , then .

Rolle's Theorem.

If a function satisfies all the following three conditions:

(1) is continuous on the closed interval ,

(2) is differentiable in the open interval ,

(3) ;

then there exists at least a point such that .

L agrange's Theorem.

If a function is

(1) continuous on the closed interval and

(2) differentiable in the open interval ,

then there exists at least a point such that

.

Limits- indeterminate forms and L’Hospital’s Rule

1. Indeterminate Form of the Type

We have previously studied limits with the indeterminate form as shown in the following example:

Example:

However, there is a general, systematic method for determining limits with the indeterminate form . Suppose that f and g are differentiable functions at x = a and that is an indeterminate form of the type ; that is, and . Since f and g are differentiable functions at x = a, then f and g are continuous at x = a; that is, = 0 and = 0. Furthermore, since f and g are differentiable functions at x = a, then and . Thus, if , then

if and

are continuous at x = a. This illustrates a special case of the technique known as L’Hospital’s Rule.