FINAL PRESENTATION

INSTRUCTOR: MISS RABIA JAVED

BASIC MATHMETICS

INTRODUCTION

GROUP LEADER : MR.ADEEL IFTIKHAR(20397)

GROUP MEMBER : MR.FAIZAN FARAZ(20274) MR.ABDUL HASEEB(20272)

AGENDA FOR TODAY DEFINITION OF DERIVATIVE DERIVATIVE NOTATIONS DERIVATIVE RULES DERIVATIVE OF CONSTANT WITH

EXAMPLE POWER RULE WITH EXAMPLE PRODUCT RULE WITH EXAMPLE QOUTIENT RULE WITH EXAMPLE

DERIVATIVE DEFINITION

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). It is a fundamental tool of variable.

EXAMPLE OF DERIVATIVE

The derivative of the position of a moving object with respect to time is the object's velocity

DERIVATIVE NOTATIONS

These are the notations of derivative. F’(x) by Lagrange dy/dx by Leibniz Dx f(x) or Dxy by Euler Ϋ by Newton 

RULES OF DERIVATIVE

These are the rules of derivatives. The derivative of constant f(x) is always Zero. The derivative of x is always 1. The Power Rule is subdivided into two parts1. If f(x) = xⁿ if n is positive then f’(x)= nxn-1

2. If f(x) = xn if n is negative thenf’(x)= nxn-1

3. If f(x) = xn if n is fraction thenf’(x)= nxn-1

CONTINUED

Rules for derivation of addition & subtraction of functions.

1. If f(x) + g(x)Then their derivative would be like this f’(x) + g’(x)2. If f(x) – g(x)Then their derivative would be like this f’(x) – g’(x)

CONTINUED

Product Rule let f(x).g(x)Then their derivative would be like

= f’(x).g(x) + g’(x).f(x)

CONTINUED

The last rule is Quotient Rule.let suppose h(x) = f(x)/g(x)

Then their derivative would like dh(x)/dx = { f’(x).g(x) – g’(x).f(x) }/g(x)2

Example of Constant functions

let f(x) = -55As -55 is a constant number so applying That derivative of constant function is always zero so

f’(x) = 0

Example of Constant functions

Let f(x) = 4x0

As we know that any number or variable raised to power zero is 1.So x0 = 1Then f(x) = 4.1 = 4 And f(x) = 4

Continued

So by derivative of constant function is always zero.

f’(x) = 0

ADBUL HASEEB

ID = 20272

Number of DerivationsThere are also represented by 1st derivative ,2nd derivatives it mean by that how many times you derivate the functions.The notations F’(x) as 1st derivative.F’’(x) as 2nd derivative.F’’’(x) as 3rd derivative.F’’’’(x) as 4th derivative.

Derivative in terms of limitDerivative can also be represented in terms of limit as

And limit can be defined as x approaches to c there is a value L.

Examples of derivative of x1

let f(z) = 4zSo using that derivative of z1 = zSo f(z) = 4.z f(z) = 4zTaking derivative on both sides f’(z) = 4

Example of derivative of x1

Let f(x) = -3x/4+9 As 4+9 = 13 f(x) = -3x/13As we know derivative of x1 = 1But firstly taking derivative on both sides f’(x) = -3.(dx/dx)/13So f’(x) = -3/13

Examples of Power Rule

Let f(x) = x10

Taking derivative on both sidesAs power is positive so xn = nxn-1

So f’(x) = 10x10-1

f’(x) = 10x9

Examples of Power Rule

Let f(x) = -10/x4

Shifting x4 upwards then function will become

f(x) = -10x-4

As power is negative so derivative will be f(x) = xn f’(x) = nxn-1

Continued

Taking derivative on both sidesf’(x) = -10.(-4.x-4-1)f’(x) = +40.x-5

So the function in the end will look like

f’(x) = 40/x5

Example of Power Rule

Let f(x) = √x5 As in powers (xm)n = xm.n

So the function will become f(x) = x5/2

By power rule f(x) = xa/b => f’(x) = (a/b)xa/b-1

f’(x) = (a/b)x(a-b)/b

Continued

So by applying thef’(x) = (5/2)x5/2 – 1

f’(x) = (5/2)x(5-2)/2 As 5/2 = 2.5f’(x) = (2.5)x3/2

f’(x) = 2.5x3/2

ADEEL IFTIKHAR

ID :- 20397

Continuity & Differentiability

For derivation it must be kept in mind that for derivation a function must be continues. Such that

Left hand limit = Right hand limit

Derivative Functions

A function whose domain remains same after derivation. Examplef(x) = x3 if 1 < x < 10And its derivative function will the same limitf’(x) = 3x2 1 < x < 10

Examples of Product Rule

Let f(x) = (x3 – 2x)(x5 + 6x2)Using power rule of derivative which isf(x) = g(x).h(x)Then f’(x) = g’(x).h(x) + h’(x).g(x)

Continued

By apply Power Rulef’(x) = d(x3 – 2x)/dx.(x5 + 6x2) +

d(x5 + 6x2)/dx.(x3 – 2x)f’(x) = (dx3/dx – d2x/dx).(x5 + 6x2) +

(dx5/dx + d6x2/dx.(x3 – 2x )f’(x)= {3x2(dx/dx) - 2 (dx/dx)}.(x5 + 6x2) +

{5x4(dx/dx) + 12x(dx/dx)}.(x3 – 2x )

Continued

f’(x)= (3x2 – 2).(x5 + 6x2) + (5x4 + 12x).(x3 – 2x)

f‘(x) = {3x2 (x5 + 6x2) -2(x5 + 6x2)} + {5x4 (x3 – 2x) + 12x(x3 – 2x)}

f’(x) = (3x2.x5 + 18x2.x2 - 2x5 - 12x2 ) + (5x4.x3 – 10x4.x + 12x.x3 – 24x.x)

Continued

f’(x) = (3x2.x5 + 18x2.x2 - 2x5 - 12x2 ) + (5x4.x3 – 10x4.x + 12x.x3 – 24x.x)

f’(x) = (3x2+5 + 18x2+2 - 2x5 - 12x2 ) + ( 5x4+3 – 10x4+1 + 12x1+3 – 24x1+1)

f’(x) = 3x7 + 18x4 - 2x5 - 12x2 + 5x7 – 10x5 + 12x4 – 24x2

Continued

f’(x) = 3x7 + 18x4 - 2x5 - 12x2 + 5x7 – 10x5 + 12x4 – 24x2

f’(x) = 8x7 + 30x4 - 12x5 - 36x2

Example of Quotient rule

Let f(x) = (x+2)/x3

Using Quotient rule which is h(x) = f(x)/g(x)Then their derivative will be=> h’(x) = { f’(x).g(x) – g’(x).f(x) } / [f(x)]2

Continued

By applying Quotient Rule f’(x) = { [d(x+2)/dx].x3 – [d(x3)/dx].(x+2) } / (x3)2

f’(x)=[dx/dx + d2/dx].x3 -[3x2.dx/dx].(x+2)/x6

f’(x) = { (1 + 0). x3 - 3x2.(x+2) }/x6

f’(x) = { (1) x3 -3x3 + 6x2}/x6

Continued

Simplifying the answer

=> f’(x) = { x3 -3x3 + 6x2}/x6

Þ f’(x) = {-2x3 + 6x2}/x6

Þ f’(x) = x2.{-2x1 + 6}/x6

Þ f’(x) = {-2x1 + 6}/x6-2

Þ f’(x) = (-2x1 + 6)/x4

So the derivative isf’(x) = (-2x1 + 6)/x4

ANY QUUSTIONS

THANK YOU