12 differentiation

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation

M. Norazizi Sham Mohd Sayuti, PhD

Faculty Science and Technology

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

2

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of the sloping straight line in the figure is defined as:

the vertical distance the line rises and falls between the two points P and Qthe horizontal distance between P and Q

3

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

5

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a curve at a given point

If we take two points P and Q on a curve and calculate, as we did for the straight line, the ratio of the vertical distance the curve rises or falls and the horizontal distance between P and Q the result will depend on the points chosen:

6

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a curve at a point P is defined to be the gradient of the tangent at that point:

The gradient of a curve at a given point

8

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

9

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Algebraic determination of the gradient of a curve

The gradient of the chord PQ is and the gradient of the tangent at P is y

x

dy

dx

10

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Algebraic determination of the gradient of a curve

As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.

For example, consider the graph of 22 5y x=

11

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Algebraic determination of the gradient of a curve

At Q:

So

As

Therefore

called the derivative of y with respect to x.

( )2

2 5y y x x =

222 4 . 2 5x x x x =

2

4 . 2 and 4 2.y

y x x x x xx

= =

0 so the gradient of the tangent at y dy

x Px dx

=

4dy

xdx

=

12

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

13

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Derivatives of powers of x

Two straight lines

(a) (constant)y c=

0 therefore 0dy

dydx

= =

14

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Derivatives of powers of x

Two straight lines

(b) y ax=

. therefore dy

dy a dx adx

= =

( ) y dy a x dx =

15

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Derivatives of powers of x

Two curves

(a)

so

2 y x=

therefore 2dy

xdx

=

2( ) y y x x =

2

2 . therefore 2y

y x x x x xx

= =

16

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Derivatives of powers of x

A clear pattern is emerging:

1If then n ndyy x nx

dx-= =

18

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

19

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation of polynomials

To differentiate a polynomial, we differentiate each term in turn:

4 3 2

3 2

3 2

If 5 4 7 2

then 4 5 3 4 2 7 1 0

Therefore 4 15 8 7

y x x x x

dyx x x

dx

dyx x x

dx

= - -

= - -

= -

20

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

24

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Standard derivatives and rules

Standard derivatives

The table of standard derivatives can be extended to include trigonometric and the exponential functions:

d

dxsin x( ) = cos x

d

dxcos x( ) = -sin x

d

dxex( ) = ex

27

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

28

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation of products of functions

Given the product of functions of x:

then:

This is called the product rule.

y uv=

dy dv duu v

dx dx dx=

29

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation of products of functions

If:

then

For example:

y uv

dy dv duu v

dx dx dx

=

=

( )

3

3 2

2

.sin 3 then

.3cos3 3 sin 3

3 cos3 sin 3

y x x

dyx x x x

dx

x x x x

=

=

=

30

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

31

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation of a quotient of two functions

Given the quotient of functions of x:

then:

This is called the quotient rule.

uy

v=

2

du dvv u

dy dx dxdx v

-=

32

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Differentiation of a quotient of two functions

If:

then

For example:

2

u

yv

du dvv u

dy dx dxdx v

=

-=

2

sin 3 then

1

( 1)3cos3 sin 3 .1

( 1)

xy

x

dy x x x

dx x

=

-=

33

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

The gradient of a straight-line graph

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Differentiation of polynomials

Derivatives – an alternative notation

Standard derivatives and rules

Differentiation of products of functions

Differentiation of a quotient of two functions

Functions of a function

34

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Functions of a function

Differentiation of a function of a function

To differentiate a function of a function we employ the chain rule.

If y is a function of u which is itself a function of x so that:

Then:

This is called the chain rule.

( ) ( [ ])y x y u x=

dy dy du

dx du dx=

35

STROUD Worked examples and exercises are in the text

KEH

1123 E

ng

ineeri

ng

Math

em

ati

cs I

Functions of a function

If:

then

For example:

( ) and ( )

. (called the 'chain rule')

y f u u F x

dy dy du

dx du dx

= =

=

cos(5 4) so cos and 5 4

.

( sin ).5

5sin(5 4)

y x y u u x

dy dy du

dx du dx

u

x

= - = = -

=

= -

= - -36