DIFFERENTIATION
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 =
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
Therefore
=
The gradient at A is
=
or
This is known as differentiating by first principle
From the first principle
i) f(x)= x
ii) f(x)= x2
=
∴
iii) f(x) =x3
iv) f(x)= xn
By binomial series
In general
If
Example
Differentiate the following with respect to x
i)y = x2+3x
Solution
y =x2+3x
ii) 2x4+5
iii)
=
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
Therefore
Therefore
If y= uv
Therefore
If y= uv
It is the product rule
Examples
Differentiate the following with respect to x
i) y = [ x2+3x] [4x+3]
ii) y = [ +2] [x2+2]
Solution
Y = [x2+3x] [4x+3]
Let u = x2+3x
= 2x+3
V = 4x+3
Therefore
=4x2+12x+8x2+12x+6x+9
=12x2+30x+9
ii)Let u = +2 →
ii)Let u = +2 →
v = x2+2 =2x
Therefore
DIFFERENTIATION OF A QUOTIENT [QUOTIENT RULE]
Let y = where u and v are functions of x
As
Exercise
Differentiate the following with respect to x
I.
II.
DIFFERENTIATION OF A FUNCTION [CHAIN RULE]
If y = f(u), where u = f(x)
Then
Therefore
PARAMETRIC EQUATIONS
Let y = f(t) , and x = g (t)
Example
I. Find if y = at2 and x = 2at
Solution
2at
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.
∴
2. x2 +y2 + 2xy = 5
Exercise
Find when x3 + y3 – 3xy2 = 8
Differentiation of trigonometric functions
1) Let y = sin x….. i
…(ii
,
Provided that x is measured in radian [small angle]
2. Let y = cos x …… (i)
3. Let
From the quotient rule
4. Let y = cot x
∴
5. y =
∴
Let y =cosec x
Therefore
Differentiation of inverses
1) Let y = sin-1x
x = sin y
2) Let y = cos-1 x
x= cos y
3) Let y = tan -1 x
x = tan y
Let y =
X =
∴
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)
Differentiation of logarithmic and exponential functions
1- Let y = ln x
Example
Find the derivative of
Solution
By quotient rule
2- Let y =
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
2) Let
Since â„®x does not depend on h,then
Therefore
Example
Find the derivative of y = 105x
Solution
Solution
Y = 105x
Iny = In105x
Therefore
Exercise
Find the derivatives of the following functions
a) a) Y =
b) b) Y =
c) c)Y=
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
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
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
Volume of the cone
Volume of the cone
V
The ratio of corresponding sides
Given = 20cm3/s
V=
So ,
Then,
A 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
Volume of horse trough
From the ratio of the corresponding sides
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
TANGENTS AND NORMALS
From a curve we can find the equations of the tangent and the normal
Example
i. Find the equations of the tangents to the curve y =2x2 +x-6 when x=3
Solution
(x. y)= (3, 5) is the point of contact of the curve with the tangent
But
Gradient of the tangent at the curve is
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
Slope of the tangent [m] = -3
Equation of the tangent at (0, 2) is
Slope of the normal
From; m1m2 = -1,Given m1=-3
Equation of the normal is
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 = 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 changes from being negative to positive i.e.
Increasing as x- increases
Is positive at the minimum point
Is positive for minimum value of the function of (y)
Maximum points
At maximum period the gradient changes from positive to negative
i.e.
Decreases as x- increases
Is negative at the maximum value of the function (y)
Point of inflection
This is the changes of the gradient from positive to positive
Is positive just to the left and just to the left
This is changes of the gradient from negative to negative.
Is negative just to the left and just to the right
Is zero for a point of inflection i.e
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
At stationary points
Therefore,
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)
Point (0,-6) is a point of inflection
∙ At (-3,-33)
At (-3, -33) is a minimum point
Alternatively,
You test 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 = +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 = 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
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
∙ 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
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
Putting the expressions a1,a2,a3,a4,a5,………back to the original series and get
which is the maclaurin series.
Examples
Expand the following
i) â„®x
ii) f(x) = cos x
Solution
i.
ii.
Exercise
∙ Write down the expansion of
∙ If x is so small that x3 and higher powers of x may be neglected, show that
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
Example
∙ Expand in ascending powers of h up to the h3 term, taking as
1.7321 And 5.50 as 0.09599c find the value of cos 54.5 to three decimal places
Solution
∙ Obtain the expansion of 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)
Keeping x constant and differentiate f with respect to y
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 by
f(x, y) = sin(xy) +cos x
Solution
Exercise
1. find fx and fy if f(x,y) is given by
a)
b)
c)
d)
Suppose compute
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