Home ADVANCED LEVEL TOPIC 9: DIFFERENTIATION | MATHEMATICS FORM 5

TOPIC 9: DIFFERENTIATION | MATHEMATICS FORM 5

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DERIVATIVES

Slope of a curve

A curve has different slopes at each point. Let A, B, and C be different points of  a curve f (x)

Where ðx is the small increase in x

ðy is the small increase in y

The slope of chord AC = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image002.gif

If C moves right up to A the chord AC becomes the tangent to the curve at A and the slope at A is the limiting value of    E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image002.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\del.png

Therefore

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image002.gif  = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image005.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image006.gif

The gradient at A is

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image007.gif = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image008.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\delta.png
or

E:\..\..\..\thlb\cr\tz\__I__Images__I__\h28.png

This is known as differentiating by first principle

From the first principle

i)                   f(x)= x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image011.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image012.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\120.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\215.png

ii)                f(x)= x2

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image015.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image016.gif

=  E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image017.gif

∴ E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image018.gif

iii)           f(x) =x3

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image019.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image020.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image021.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image022.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image023.gif

iv)    f(x)= xn

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image024.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image025.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image026.gif

By binomial series

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image028.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image029.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image030.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image031.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image032.gif

In general

If E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image033.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\save.png

Example

Differentiate the following with respect to x

i)y = x2+3x

Solution

y =x2+3x
E:\..\..\..\thlb\cr\tz\__I__Images__I__\diff.png

ii)    2x4+5
E:\..\..\..\thlb\cr\tz\__I__Images__I__\cube.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\rea.png
iii) E:\..\..\..\thlb\cr\tz\__I__Images__I__\34Ea.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image002.gif = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image037.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\756Rt1.png

Differentiation of products functions [ product rule]

Let y =uv

Where u and v are functions of x

If x   →   x+ðx

u     →    u +ðu

v     →   v+ðv

y     →    ðy +y

y= uv ……i)

Therefore

y+ðy = [ u+ðu][v+ðv]

y+ðy = uv +uðv+vðu +ðuðv….ii

Subtract (i) from (ii)

δy  =uðv +vðu +ðuðv

Therefore

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image039.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\ja.png

Therefore
E:\..\..\..\thlb\cr\tz\__I__Images__I__\captured.png
Therefore
If y= uv

E:\..\..\..\thlb\cr\tz\__I__Images__I__\one80.png     It is the product rule

Examples

Differentiate the following with respect to x

i)    y = [ x2+3x] [4x+3]

ii)    y = [ E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image044.gif  +2] [x2+2]

Solution

Y = [x2+3x] [4x+3]

Let u = x2+3x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image045.gif  = 2x+3

V = 4x+3

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image046.gif

Therefore
E:\..\..\..\thlb\cr\tz\__I__Images__I__\diff1.png

=4x2+12x+8x2+12x+6x+9

=12x2+30x+9
E:\..\..\..\thlb\cr\tz\__I__Images__I__\twelz.png
ii)Let u =  E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image044.gif +2     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image048.gif   → E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image049.gif

v = x2+2   E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image034.gif    E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image050.gif  =2x

Therefore

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image051.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image052.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\good.png

DIFFERENTIATION OF A QUOTIENT [QUOTIENT RULE]

Let y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image054.gif  where u and v are functions of x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image055.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image056.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image057.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image058.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image059.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image060.gif

As E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image061.gif
E:\..\..\..\thlb\cr\tz\__I__Images__I__\ja.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\w2.png

Exercise

Differentiate the following with respect to x

I.    E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image063.gif

II.     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image064.gif

DIFFERENTIATION  OF A FUNCTION [CHAIN RULE]

If y = f(u), where u = f(x)

Then

E:\..\..\..\thlb\cr\tz\__I__Images__I__\124.png

Therefore

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image066.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\216.png

PARAMETRIC EQUATIONS

Let y = f(t) , and x = g (t)

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image068.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\38.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image070.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image071.gif

Example

I.            Find E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif  if y = at2 and x = 2at

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image073.gif 2at

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image074.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image075.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image076.gif

IMPLICIT  FUNCTION

Implicit function is the one which is neither x nor y a subject e.g.

1)     x2+y2 = 25

2)     x2+y2+2xy=5

One thing to remember is that y is the function of x

Then

1.                E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image077.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\55.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image079.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image080.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image081.gif

∴     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image082.gif

2.     x2 +y2 + 2xy = 5

E:\..\..\..\thlb\cr\tz\__I__Images__I__\46.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image084.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image085.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image086.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image087.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\dy1.png

Exercise

Find E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif when x+ y3 – 3xy2 = 8

Differentiation of trigonometric functions

1)     Let y = sin x….. i

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image089.gif …(ii E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image090.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image091.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image092.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image093.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image094.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image095.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\126.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image097.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image098.gif , E:\..\..\..\thlb\cr\tz\__I__Images__I__\217.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\39.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\derivative.png

Provided that x is measured in radian [small angle]

2. Let y = cos x …… (i)

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image101.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\gives.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image102.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\56.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\561.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image105.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\delta1.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image107.gif

3.      Let E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image108.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image109.gif

From the quotient rule

E:\..\..\..\thlb\cr\tz\__I__Images__I__\qaas.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image111.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image112.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image113.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\w3.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\as3.png

4.      Let y = cot x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image115.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image116.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\131.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\224.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\312.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\132.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\225.png

∴ E:\..\..\..\thlb\cr\tz\__I__Images__I__\313.png

5.      y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image123.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image124.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image125.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image126.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\133.png

∴ E:\..\..\..\thlb\cr\tz\__I__Images__I__\134.png

Let y =cosec x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image129.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\135.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image131.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image132.gif

Therefore           E:\..\..\..\thlb\cr\tz\__I__Images__I__\1211.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\136.png

Differentiation of inverses

1)     Let y = sin-1x

x = sin y

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image135.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image136.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image137.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image138.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\1212.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\314.png

2)     Let y = cos-1 x

x= cos y

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image141.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image142.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image143.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image144.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image145.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\1213.png

3)     Let y = tan -1 x

x = tan y

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image146.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image147.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\137.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\2110.png

Let y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image150.gif

X = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image151.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image152.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image153.gif

∴ E:\..\..\..\thlb\cr\tz\__I__Images__I__\34E.png

Exercises

        Differentiate the following with respect to x.

i)   Sin 6x

ii)   Cos (4x2+5)

iii)   Sec x tan 2 x

        Differentiate sin2 (2x+4) with respect to x

        Differentiate the following from first principle

i)    Tan x

ii)   E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image155.gif

Differentiation of logarithmic and exponential functions

1-     Let y = ln x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image156.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image157.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image158.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image159.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image160.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image161.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image162.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image163.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image164.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\qw1.png

Example

Find the derivative of E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image167.gif

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image168.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image169.gif

By quotient rule

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image170.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\rfwe.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image172.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image173.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image174.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image175.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\qw2.png

2-     Let y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image177.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\1231.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image179.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image180.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image181.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image182.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image183.gif

Differentiation of Exponents

1)     Let y = ax

If a function is in exponential form apply natural logarithms on both sides

i.e.  ln y = ln ax

ln y =x ln a

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image184.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image185.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image186.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\urr.png

2)     Let E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image187.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image188.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image189.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image190.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image191.gif

Since â„®x does not depend on h,then

E:\..\..\..\thlb\cr\tz\__I__Images__I__\1232.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image193.gif

Therefore

E:\..\..\..\thlb\cr\tz\__I__Images__I__\as4.png

Example

Find the derivative of y = 105x
Solution

Y = 105x

Iny = In105x

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image197.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\t3R4E.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image199.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image200.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image201.gif

Therefore

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image202.gif

Exercise

Find the derivatives of the following functions

a)  a)   Y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image203.gif

b)    b) Y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image204.gif

c)     c)Y= E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image205.gif

APPLICATION OF DIFFERENTIATION

Differentiation is applied when finding the rates of change, tangent of a curve, maximum and minimum etc

i) The rate of change

Example

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image034.gif The side of a cube is increasing at the rate of 6cm/s. find the rate of increase of the volume when the length of a side is 9cm

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image206.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image207.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\eg.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image209.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\wsd.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image211.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image212.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image034.gif A hollow right circular cone is held vertex down wards beneath a tap leaking at the rate of 2cm3/s. find the rate of rise of the water level when the depth is 6cm given that the height of the cone is 18cm and its radius is 12cm.

Solution
E:\..\..\..\thlb\cr\tz\__I__Images__I__\aw7.jpg

Volume of the cone

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image220.gif

The ratio of corresponding sides E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image221.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image222.gif

Given E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image223.gif = 20cm3/s E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image224.gif

V= E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image225.gif

So , E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image226.gif

Then, E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image227.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image228.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image229.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image230.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image231.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image232.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image034.gifA horse trough has triangular cross section of height 25cm and base 30cm and is 2m long. A horse is drinking steadily and when the water level is 5cm below the top is being lowered at the rate of 1cm/min find the rate of consumption in litres per minute

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image233.jpg

Volume of horse trough

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image234.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image235.gif

From the ratio of the corresponding sides

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image236.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image237.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image238.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image239.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image240.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image241.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image242.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image243.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image244.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image245.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image246.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image247.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image248.gif   E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image249.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image034.gif A rectangle is twice as long as it is broad find the rate change of the perimeter when the width of the rectangle is 1m and its area is changing at the rate of 18cm2/s assuming the expansion is uniform

Solution

E:\..\..\..\thlb\cr\tz\__I__Images__I__\a37.jpg

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image250.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image251.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image252.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image253.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image254.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image255.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image256.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image257.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image258.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image259.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image260.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image261.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image262.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image263.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image264.gif

TANGENTS AND NORMALS

From a curve we can find the equations of the tangent and the normal

E:\..\..\..\thlb\cr\tz\__I__Images__I__\a38.jpg

Example

i.            Find the equations of the tangents to the curve  y =2x2 +x-6 when x=3

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image270.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image271.gif

(x. y)= (3, 5) is the point of contact of the curve with the tangent

But

Gradient of the tangent at the curve is

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image272.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image273.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image274.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image275.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image276.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image277.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image278.gif

Example

ii.            Find the equation of the tangent and normal to the curve y = x2 – 3x + 2 at the point where it cuts y axis

Solution

The curve cuts y – axis when x = 0

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image279.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image280.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image281.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image282.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image283.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image284.gif

Slope of the tangent [m] = -3

Equation of the tangent at (0, 2) is

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image285.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image286.gif

Slope of the normal

From; m1m2 = -1,Given m1=-3

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image287.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image288.gif

Equation of the normal is

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image289.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image290.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\324.png

Exercise

  Find the equation of the tangent to 2x2 – 3x which has a gradient of 1

  Find the equations of the normal  to the curve  y = x2-5x +6 at the points where the curve cuts the x axis

Stationary points [turning points]

A stationary point is the one where by E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif  = 0 it involves:

        Minimum turning point

        Maximum turning point

        Point of inflection

Nature of the curve of the function

At point A, a maximum value of a function occurs

At point B, a minimum value of a function occurs

At point C, a point of inflection occurs

At the point of inflection is a form of S bend

Note that

Points A, B and C are called turning points on the graph or stationary values of the function

Investigating the nature of the turning point

Minimum points

At turning points the gradient E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif   changes from being negative to positive i.e.

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image299.gif Increasing as x- increases

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image300.gif  Is positive at the minimum point

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image301.gif   Is positive for minimum value of the function of (y)

Maximum points

At maximum period the gradient changes from positive to negative

i.e.

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image303.gif  Decreases as x- increases

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image301.gif  Is negative at the maximum value of the function (y)

Point of inflection 

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image304.jpg

This is the changes of the gradient from positive to positive

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image303.gif  Is positive just to the left and just to the left

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image305.jpg

This is changes of the gradient from negative to negative.

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image303.gif  Is negative just to the left and just to the right

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image300.gif  Is zero for a point of inflection i.e

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image306.gif   Is zero for point of inflection

Examples

Find the stationary points of the and state the nature of these points of the following functions

Y = x4 +4x3-6

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image307.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image308.gif

At stationary points E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image309.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image310.gif

Therefore, E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image311.gif

Then the value of a function

At x = 0, y = -6

X= -3, y =-33

Stationary point at (0,-6) and (-3,-33)

At (0,-6)

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image312.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image313.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image314.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image315.gif

Point (0,-6) is a point of inflection

        At (-3,-33)

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image316.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image317.gif

At (-3, -33) is a minimum point

Alternatively,

You test  E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif   by taking values of x just to the right and left of the turning point

Exercise

1)     1. Find and classify the stationary points of the following curves

a)     (i) y = 2x-x2

b)     (ii) y = E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image318.gif  +x

c)     (iii) y= x2(x2– 8x)

2)    2. Determine the smallest positive value of x at which a point of inflection occurs on the graph of y = 3â„®2x cos (2x-3)

3)    3. If 4x2 + 8xy +9y2 8x – 24y +4 =0 show that when E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image072.gif = 0,

x + y = 1. Hence find the maximum and minimum values of y

Example

     1. A farmer has 100m of metal railing with which to form two adjacent sides of a rectangular enclosure, the other two sides being two existing walls of the yard meeting at right angles, what dimensions will give  the maximum possible area?

Solution       

E:\..\..\..\thlb\cr\tz\__I__Images__I__\a40.jpg

Where, W is the width of the new wall

L is the length of the new wall

The length of the metal railing is 100m

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image320.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image321.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image322.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image323.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image324.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image325.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image326.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image327.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image328.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image329.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image330.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image331.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image332.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image333.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image334.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image335.gif

      2. An open  card board box width a square base is required to hold 108cm3 what should be the dimensions if the area of cardboard used is as small as possible

Solution

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image341.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image342.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image343.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image344.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image345.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image347.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image349.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image350.gif

Differentiation

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image353.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image355.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image357.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image358.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image359.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image360.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image361.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image362.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image363.gif

Exercise

The gradient function of y = ax2 +bx +c is 4x+2. The function has a maximum value of 1, find the values of a, b, and c

MACLAURIN’S SERIES [from power series ]

Let f(x) = a1 +a2x+a3x2 +a4x3 +a5x4+ a6x5…….i

In order to establish the series we have to find the values of the constant co efficient a1, a2, a3, a4, a5, a6 etc

Put x = 0 in …i

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image364.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image365.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image367.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image368.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image370.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image371.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image372.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image373.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image375.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image376.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image377.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image378.gif

Differentiation

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image381.gif

Differentiation

Putting the expressions a1,a2,a3,a4,a5,………back to the original series and get

Differentiation which is the maclaurin series.

Examples

Expand the following

i)    â„®x

ii)  f(x) = cos x

Solution

i.      Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image386.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image387.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image389.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image391.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image392.gif

ii.                        E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image393.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image394.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image395.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image396.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image397.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image398.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image399.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image400.gif

Exercise

        Write down the expansion of  E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image401.gif

        If x is so small that x3 and higher powers of x may be neglected, show that E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image402.gif

TAYLOR’S SERIES

Taylor’s series is an expansion useful for finding an approximation for f(x) when x is close to a

By expanding f(x) as a series of ascending powers of (x-a)

f(x) = a0 +a1(x-a) +a2(x-a)2+a3(x-a)3 +……..

This becomes

Differentiation

Example

        Expand E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image404.gif in ascending powers of h up to the h3 term, taking E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image405.gif as

1.7321 And 5.50 as 0.09599c find the value of cos 54.5 to three decimal places

Solution

Differentiation

Differentiation

Differentiation

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image410.gif

Differentiation

Differentiation

Differentiation

Differentiation

        Obtain the expansion of E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image415.gif in ascending powers of x as far as the x3term

Introduction to partial derivative

Let f (x, y) be a differentiable function of two variables. If y kept constant and differentiates f (assuming f is differentiable with respect to x)

Differentiation

Keeping x constant and differentiate f with respect to y

Differentiation

Example

        find the partial derivatives of fx and fy

If f(x, y) = x2y +2x+y

Solution

Differentiation

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image420.gif

Differentiation

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image422.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\ewq.png

        Find fx and fy if f (x,y) is given by

f(x, y) = sin(xy) +cos x

Solution

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image424.gif

E:\..\..\..\thlb\cr\tz\__I__Images__I__\qw.png

E:\..\..\..\thlb\cr\tz\__I__Images__I__\321.png

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image427.gif

E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image428.gif

Exercise 

1.      find fx and fy if f(x,y) is given by

a)     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image429.gif

b)     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image430.gif

c)     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image431.gif

d)     E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image432.gif

Suppose E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image433.gif  compute E:\..\..\..\thlb\cr\tz\differentiationfvi_Files\image434.gif

NEXT TOPIC 

TOPIC 10: INTEGRATION | MATHEMATICS FORM 5

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