DIFFERENTIATION

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DIFFERENTIATION
DERIVATIVES
Slope of a curve
A curve has different slopes at each point. Let A, B, and C be different poin...

=
The gradient at A is
=
or
This is known as differentiatingby first principle
From the first principle
i) f(x)= x
ii) f(x...

=
∴
iii) f(x) =x3
iv) f(x)= xn
By binomial series

In general
If
Example
Differentiatethe followingwith respect to x
i)y = x2+3x
Solution
y =x2+3x
ii) 2x4+5

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iii)
=
Differentiationofproducts functions [ product rule]
Let y =uv
Where u and v are functions of x
If x → x+ðx
u → u +ð...

Therefore
Therefore
If y= uv
It is the product rule
Examples
Differentiatethe followingwith respect to x
i) y = [ x2+3x] [...

=12x2+30x+9
ii)Let u = +2 →
v = x2+2 =2x
Therefore
DIFFERENTIATION OF A QUOTIENT [QUOTIENT RULE]
Let y = where u and v are...

As
Exercise
Differentiatethe followingwith respect to x
I.
II.
DIFFERENTIATION OF A FUNCTION [CHAIN RULE]
If y = f(u), whe...

Example
I. Find if y = at2 and x = 2at
Solution
2at
IMPLICIT FUNCTION
Implicit functionis the one which is neither x nor y...

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Then
1.
∴
2. x2 +y2 + 2xy = 5
Exercise

Find when x3 + y3 – 3xy2 = 8
Differentiationoftrigonometricfunctions
1) Let y = sin x….. i
…(ii
,
Providedthat x is measur...

2. Let y = cos x …… (i)
3. Let
From the quotient rule

4. Let y = cot x
∴
5. y =

∴
Let y =cosecx
Therefore
Differentiationofinverses
1) Let y = sin-1x
x = sin y

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2) Let y = cos-1 x
x= cos y
3) Let y = tan -1 x
x = tan y

Let y =
X =
∴
Exercises
 Differentiate the followingwith respect to x.
i) Sin 6x
ii) Cos (4x2+5)
iii) Sec x tan 2 x
 Dif...

i) Tan x
ii)
Differentiationoflogarithmicandexponential functions
1- Let y = ln x
Example

Find the derivative of
Solution
By quotient rule
2- Let y =

DifferentiationofExponents
1) Let y = ax
If a functionis in exponential form apply natural logarithms on bothsides
i.e. ln...

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Since â„®x does not depend on h,then
Therefore
Example
Find the derivative of y = 105x
Solution
Y = 105x
Iny = In105x

Therefore
Exercise
Find the derivatives of the followingfunctions
a) a) Y =
b) b) Y =
c) c)Y=
APPLICATION OF DIFFERENTIATI...

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 ...

A horse trough has triangular cross section of height 25cm and base 30cm and is 2m long.
A horse is drinking steadily and ...

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 an...

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TANGENTS AND NORMALS
From a curve we can find the equations of the tangent and the normal
Example
i. Find the equations of...

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
Sol...

Equation of the normal is
Exercise
 Find the equation of the tangent to 2x2 – 3x which has a gradient of 1
 Find the equ...

Nature of the curve of the function
At point A, a maximum value of a functionoccurs
At point B, a minimum value of a funct...

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i.e.
Decreasesas x- increases
Is negative at the maximum value of the function(y)
Point ofinflection
This is the changes o...

Is negative just to the left and just to the right
Is zero for a point of inflectioni.e
Is zero for point of inflection
Ex...

Point (0,-6) is apoint of inflection
 At (-3,-33)
At (-3, -33) is aminimum point
Alternatively,
You test by taking values...

 1. A farmer has 100m of metal railing with which to form two adjacent sides of a rectangular
enclosure, the other two si...

 2. An open card board box width a square base is required to hold 108cm3 what should be
the dimensions if the areaof car...

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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, ...

MACLAURIN’S SERIES [from power series ]
Let f(x) = a1 +a2x+a3x2 +a4x3 +a5x4+ a6x5…….i
In order to establish the series we ...

Puttingthe expressionsa1,a2,a3,a4,a5,.........backtothe original seriesandget
whichisthe maclaurinseries.
Examples
Expand ...

ii.
Exercise
 Write down the expansion of
 If x is so small that x3 and higher powers of x may be neglected, show that
T...

1.7321 And 5.50 as 0.09599c findthe value of cos 54.5 to three decimal places
Solution
 Obtain the expansion of in ascend...

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Example
 find the partial derivatives of fx and fy
If f(x, y) = x2y +2x+y
Solution
 Find fx and fy if f (x,y) is given b...

Exercise
1. find fx and fy if f(x,y) is given by
a)
b)
c)
d)
Suppose compute

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