Derivatives & their Applications

Mathematics Presentations

What is derivative?• The derivative of a function of a real variable measures

the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable).

• For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced. 

What is differentiation?• The process of finding a derivative is

called differentiation.• The reverse process is called antidifferentiation.• The fundamental theorem of calculus states that

antidifferentiation is the same as integration.• Differentiation is also known as the process to find rate

of change• Derivative tells us slope of function at any point.

As it is also rate of change• The rate of change of a function is expressed as a ratio

between a change in one variable relative to a corresponding change in another.

• Rate of change is also given by limit value.

Average rate of change• The Average Rate of Change is defined as the

ratio of the difference in the function f(x) as it changes from 'a' to 'b' to the difference between 'a' and 'b'. The average rate of change is denoted as A(x).

• And is given as the formula• A(x) • The Average Rate of Change Formula calculates

the slope of a line or a curve on a given range.

Average rate of change-Ex

Calculate the average rate of change of a function, f(x) = 3x + 5 as x changes from a to b ?Solution: Given, f(x) = 3x + 5 a = 3 , b = 6Putting the valuesf(3) = 3(3) + 5 :f(6) = 3(6) + 5 f(3)=14 : f(6)=23The average rate of change is,

A(x) = 

A(x) =  = A(x) = 3

Notations for derivatives Leibniz's notation. , xLagrange's notation.

f’ f”Euler's notation.

Derivative Notation

A Function

f(x)

y

X2 + 1

Newton Form

f ’(x)

y ’

(X2 + 1)’

Leibniz Form

(f(x))

(y)

(X2 + 1)

ddx

ddx

ddx

or

or

dfdx

dfdx

Derivative Notation

Notation

There are lots of ways to denote the derivative of a function y = f(x).

f’(x) the derivative of f the derivative of f with y’ y prime respect to x.

the derivative of y the derivative of f at x with respect to x. dxdy

dxdf

)(xfdxd

Rules for derivatives

Rules for derivatives

Rules for derivatives

Rules for derivatives

Rules for derivatives

Product Rule Example

Examples

Product rule:

• This is used when differentiating a product of two functions.

Differentiate x(x² + 1)

• let u = x and v = x² + 1

Examples

Product rule:

= Putting the values.

=(x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .

Quotient Rule Example

Examples

Power Rule:

"what is the derivative of ?"• We can use the Power Rule, where n=6:

• So,• = 6 = 6

Chain rule

A special rule, the chain rule, exists for differentiating a function of another function.

In order to differentiate a function of a function, y = f(g(x)),

That is to find , we need to do two things:

1. Substitute u = g(x). This gives us y = f(u) Next we need to use a formula that is known as the Chain Rule.

2. Chain Rule

Chain Rule Example

Chain rule:If • Find Solution:

• Taking derivative to both side -------(i

• Taking derivative to both side

Chain rule:• Using chain rule

• = • = 12u

=12(2x-1) =24x-12

Chain Rule Example

Chain Rule Example

Higher derivatives. 1st ,2nd 3rd

• Any derivative beyond the first derivative can be referred to as a higher order derivative.

• The derivative of the function f(x) may be denoted by f’(x)

• Its double (or "second") derivative is denoted by f ’’(x).

• This is read as "f double prime of x," or "The second derivative of f(x)."

Higher derivatives. Notations:Let f(x) be a function in terms of x. The following are notations for higher order derivatives.

2nd Derivative Ex.

Higher Order Derivative Ex.

Partial derivatives

Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation

Partial derivatives:

The partial derivative of f is with respect to its variable.

Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee"

Example

Applications of partial derivatives:

• Derivatives are constantly used in everyday life to help measure how much something is changing. They're used by the government in population censuses, various types of sciences, and even in economics..

Applications of partial derivatives:

• Derivatives in physics.You can use derivatives a lot in Newton law of motion where the velocity is defined as the derivative of the position over time and the acceleration, the derivative of the velocity over time.

• Derivatives in chemistry.One use of derivatives in chemistry is when you want to find the concentration of an element in a product.

Concave Up

• The derivative of a function gives the slope• When the slope continually increases, the function is

concave upward.• Taking the second derivative actually tells us if the slope

continually increases or decreases.• When the second derivative is positive, the function is

concave upward.• f ''(x) > 0 for x > 0;

Concave Up:

Concave down:

• When the slope continually decreases, the function is concave downward.

• When the second derivative is negative, the function is concave downward.

• f ''(x) < 0 for x < 0.

Concave down:

Derivatives of polynomials

Derivatives of polynomials

Derivative of Trigonometric functions

Derivative of Trigonometric functions

Derivatives of Exponential and Logarithmic Functions

Derivatives of Exponential and Logarithmic Functions

• Thank You So much for Listening• Any Question?