Chapter: 6 Derivative and its Applications

Derivatives of Functions

The derivative of a function 31 is another function 31, defined by

31

The derivative at some specific point 31 is a number defined by

31

Equivalent symbols:

31

31

For example, 31,

31 31 31 31 31

Tangent Line

The slope of the tangent line to the graph of 31 at the point 31 is equal 31, the equation of this tangent line is

31

Example: Given graph of the function 31, the slope of the tangent line at 31 is

31

and the equation of the tangent line is

31

Average and Instantaneous Velocity of a Moving Particle

If a particle moves along the 31-axis in such a way that its position at time 31 is 31, then its average velocity on some time interval is

31

and velocity at any time 31 is the derivative 31

Example: The motion of a particle is given by the function 31. The average velocity on the time interval 31 is

31

the velocity at instant time 31 is 31

Some Important Derivatives

Using the definition of the derivative and the properties of the limit operation the following derivatives can be calculated as:

• 31 (the derivative of a constant function is zero.)

• 31

• 31

• 31

• 31 31

• 31 31

• 31, 31

• 31

• 31 31

The Derivative Rules

31

31

the product rule:

31

31

the quotient rule

31

For example,

31

31 31

31

31 31

The Chain Rule

In order to calculate the derivatives of more complicated functions the following chain rule can be applied.

If 31 and 31 are differentiable then the composite function 31 is also differentiable and

31

Here 31 is the derivative of 31 calculated at the point 31.

For example,

31 31

31 31

31 31 31 31

31

Higher Order Derivatives

If a function 31 is differentiable, its derivative 31 is a function of 31. The derivative of the function 31 is called the second order derivative of 31 and is denoted by 31 or 31. If the function 31 is differentiable, the derivative of the function 31 is called the third order derivative of 31 and is denoted by 31 or 31. In the general, 31th order derivative is denoted by 31 or 31 provided that the derivative exists.

Examples:

31, 31, 31, 31.

31

31

31

31

31

31

31

Applications of Derivatives

Find the equation of the tanget line to the graph of the function 31 at the point 31.

31

31

the equation of the tangent line

31

Local Extremums and Monotonicity

Given 31, if 31 then 31 is called a critical point.

If for all 31 the inequality 31 is satisfied then 31 is monotone increasing on 31; if for all 31 the inequality 31 is satisfied then 31 is monotone decreasing on 31.

Example: 31. 31.

31

is a critical point,

31

that is on the interval 31 the function 31 is monotone increasing,

31

on the interval 31 the function 31 is monotone decreasing.

Second Derivative Test for Local Extrema:

If 31 is a critical point and the second derivative 31 is positive,i.e. 31, then 31 is a local minimum point.

If 31 is negative, i.e. 31, then 31 is a local maxmimum point.

Example: 31,

31,

31.

Critical points:

31

Interval of increase:

31

Interval of decrease:

31

31 31 31 is a local maximum pont.

31 31 31 is a local minimum pont.

First Derivative Test for Local Extrema:

If 31 is a critical point and 31 to the left of 31 and 31 to the right of 31 then 31 is a local minimum.

If 31 to the left of 31 and 31 to the right of 31 then 31 is a local maximum.

Example: 31, 31.

31 31 31 31 31 is a critical point.

Take 31 then 31, take 31, then 31. Therefore 31 is a local maximum.