DIFFERENTIATION

The process of finding the differential coefficient of a function f (x) with respect to the independent variable x is called differentiation.

The differential coefficient of a function is also called derived function or derivative of the function.

Let y = f (x) be a continuous function of x then , if it exist is called the derivative of f (x) or differential coefficient with respect to x

The notation which are commonly used for derivative of a function are

It is important to know that is either the gradient function of the curve y = f (x) or the rate of change of y with respect to x.

DIFFERNTIATION FROM THE FIRST PRINCIPLE

Step to follow when differentiating from the first principle

1. Put the function to be differentiated equal to y i.e. Let

2. Let be an increment in the value of x and , the corresponding increment in the

value of y as a result of an increment in x so that

3. Subtract (1) from (2) so as to obtain

4. Divide both sides by , so as to obtain i.e

5. Take the limits of both sides as

This technique of finding differential coefficient by taken into consideration of its limiting value is called differentiation from the first principle

EXAMPLE 1

Differentiate the following function with respect to x using first principle.

SOLUTION

EXAMPLE 2

Differentiate with respect to x fro the first principle

SOLUTION

DERIVATION OF xⁿ

Hence .

The relation is true for integral and fractional values of n

EXAMPLE 3

Find the derivative if each of the following

(a) (b) (c) (d)

SOLUTION

DERIVATIVE OF

EXAMPLE 4

Find the derivative of the following

SOLUTION

GENERAL THEOREM OF DIFFERENTIATION

Theorem 1: The derivative of a constant is zero that is , where c is a constant.

Let y = c

From the first principle

Hence the derivative of a constant is zero

THEOREM 2: The additive constant disappear in differentiation

EXAMPLE 5 Find the derivative of the following

SOLUTION

THEOREM 3: The derivative of the product of a constant and a function is equal to the product of constant and the derivative of the function i.e.

EXAMPLE 6

Determine the differential coefficient of the following

SOLUTION

THEOREM 4: The derivative of a sum of two functions y = u + v, where u and v are function of x and are differentiable is equal to the sum of the derivative of these functions i.e.

EXAMPLE 7

Find the derivative of the following

THEOREM 5: given a composite function also called function of a function, in which y is a function of u and that u itself is also a function x. that is y = f (u), and u = g(x) then y =f [g (x)}

The differential coefficient of y with respect to x is equal to the derivative of the first function with respect to u times the derivative of the second function with respect to .

This is called chain rule of differentiation

EXAMPLE 8

Find the derivative of each of the following

SOLUTION

THEOREM 6: The Derivative of Product

The derivative of a product y = uv, where u and v are function of x and are both differentiable functions, is equal to the product to the first function multiplied by the derivative of the second plus the second function multiplied by the derivative of the first i.e.

Proof Let y = uv

Using Differentiation from first Principle

EXAMPLE 9 Find the derivative of the following

THEOREM 7:The derivative of a Quotient

Let , where u and v are functions of x and v ¹ 0,then

Proof: Let , where u and v are functions of x .

EXAMPLE 10

Find the derivative of each of the following

IMPLICIT DIFFERENTIATION

An implicit function is a type of function in which the dependent and independent variable cannot be identified, i.e. the relation between the two variables is expressed in the form f (x, y).

For example, the equations are functions in

implicit form.

Implicit Differentiation consist of differentiating both sides of the given (defining) equation with respect to x and then solving algebraically for

Here is an example to illustrate the technique

EXAMPLE 11

SOLUTION

The trick we are going to use to differentiate y as though we are differentiating x then multiply the result with (that is derivative of y)

EXAMPLE 12

SOLUTION

EXAMPLE 13

SOLUTION

DERIVATIVE OF TRIGONOMETRIC FUNCTION

It is important to note the following

Let put all the trigonometric derivative in a table

y	dy/dx
sinx	cos x
cos x	- sin x
tan x	sec² x
cosec x	-cosec x cot x
sec x	sec x tan x
cot x	- cosec²x

Example 14

SOLUTION

EXAMPLE 15

Using differentiation from first principle find

SOLUTION

EXAMPLE 16

Find the derivative of the following

SOLUTION

EXAMPLE 17

Find the derivative of the following

Example 18

Differentiate with respect to x

THE DERIVATIVE OF LOGARTHMIC FUNCTION

EXAMPLE 19

Differentiate with respect to x

SOLUTION

EXAMPLE 20

Differentiate the following with respect to x

In the course of differentiating some function, logarithmic differentiation can be applied to reduce the complication when obtaining the derivative of the given function.

EXAMPLE 21

Use logarithmic differentiation to obtain the derivative of the following

SOLUTION

THE DERIVATIVE OF EXPONENTIAL FUNCTION

EXAMPLE 22

Find the derivative of the following

SOLUTION

Whenever, we are differentiating an exponential function of the form . The short method to obtain the derivative is given below

EXAMPLE 23

Obtain the derivative of the following

HIGHER DERIVATIVE

The entire derivative obtained so far are first order derivative. That is, given is the first order derivative

The derivative of with respect to x is is the called second order derivative or second differential coefficient which is written as pronounced dee – two – y dee x – square . The third derivative of y with respect to x is , which is denoted by .

The nth – derivative of y with respect to x is , which is denoted as . The nth derivative is obtained by calculating successive derivative in turn. For example to obtain , will first be obtained, then will be obtained from and lastly will be obtained from the differentiation of

EXAMPLE 24

SOLUTION

EXAMPLE 25

SOLUTION

DIFFERENTIATION OF INVERSE TRIG FUNCTION

In this section we shall discuss the discus the derivative of inverse trig. Functions

Find the derivative of

Solution

Worked Examples

Solution

Example

Solution

PARAMETRIC DIFFERENTIATION

Given that x and y are separately expressed as a function of a single variable say t (called parameter), then x = g (t), y = h (t) are called parametric equation

Step to differentiating parametric equation

1. Obtain and separately

2. Use

To obtain the second order derivative

a. Find

b. Use

EXAMPLE 26

Find the in terms of the parameter when

SOLUTION

EXAMPLE 27

SOLUTION

Question1

QUESTION 2

QUESTION 3

QUESTION 4

QUESTION 5

QUESTION 6

QUESTION 7

QUESTION 8

QUESTION 9

QUESTION 10

Steps When Using The Derivative To Determine The Interval Of Increase And Decrease For A Function f.

1. Determine f¹(x) from f(x)

2. Find all the value of x for which f¹(x) = 0

3. Choose a test number c from each interval a < x < b, determined in step 2 and evaluate f¹ (g) then

If f(g) > 0, the function f(x) is increasing on the interval a < x < b

If f(g) < 0, the function f(x) is decreasing on the interval a < x < b

EXAMPLE 42

Find the interval of increase and decrease for the given function

1) f(x) = x² – 4x + 5

2) f(x) = 1/3x³ – 9x + 2

SOLUTION

Interval	Test Number	f¹ (x)	Conclusion
x < - 3	- 5	f¹ (- 5)	f is increasing
- 3 < x < 3	1	f¹ (1)	f is decreasing
x > 3	4	f¹ (4)	f is increasing

MAXIMUM AND MINIMIUM VALUES RLATIVE EXTREMA

Consider the figure above with slope of the tangent marked A, B,C, D, E, F, G, H. At point C, E, H, “peaks” occurs while at point B, D, G valley occurs. There is a horizontal tangent at F that is neither a peak nor a valley.

A peak on the graph of a function f can be otherwise called relative maximum, which a valley is called relative minimum. Relative extrema is the collective term of relative maxima and minima.

The function f(x) is said to have its maximum for x = a, if the function ceases to increase and begin to decrease at x = a

The function f(x) is said to have attained its minimum value for x = b. In the figure above C, E, H are maximum point and B, D and G are minimum points. The maximum and minimum points are called turning or stationary points

The following point should be noted should

A relative maximum is a point on the graph of f that is least as high as any nearby point on the graph, while relative minimum is at least as low any nearby point.
A function may have several maximum and minimum values
Maximum and Minimum values occurs alternatively

CONDITION FOR MAXIMA AND MINIMA

At maximum point, the function y = f(x) changes from an increasing to decreasing state. That is changes from positive to negative value. must pass through the value 0. Hence = 0 at a maximum. Similarly, at minimum point y = f(x) changes from a decreasing to increasing state. changes from negative to positive. In changing from negative to positive value must pass through the value zero. Hence = 0 at a minimum point. Thus, = 0 for minimum and maximum point on the y = f(x).

Though = 0, may not necessarily be a sufficient condition for maximum or minimum point. It may happen that even if = 0, the function may go increasing or decreasing and it may not change from increasing to decreasing state or vice versa. Such a situation is described as a point of inflection. At all point where = 0 are called turning point or stationary points

CONDITION FOR MAXIMUM AND MINIMUM POINTS

Point of Inflection

Given that y = f(x), the function f(x) is said to a point of inflection for x = a if and change sign

A point of inflection may not necessarily be zero unless the tangent at the point of in figure (a) and figure (b)

Hence for the point of inflection the necessary condition are as follow

EXAMPLE 43

Find the turning points on the curve , distinguish between them. Also find the maximum and minimum value of the function

SOLUTION

SECOND METHOD

EXAMPLE 44

Find the points of inflection of the function y = 3x⁵ – 10x³

SOLUTION

EXAMPLE 45

The curve is defined , find the stationary points are maximum point and minimum point or the point of inflection

SOLUTION

Example on Minima and Maxima

EXAMPLE 46

Find two positive numbers whose sum is 50 and whose product is a large as possible

SOLUTION

EXAMPLE 47

Find two positive x and y whose sum is 30 and are such that xy² is as large possible

SOLUTION

Example 48

12 is divided into two parts such that the product of the square of one part and fourth power of the other will give a maximum. Find the two numbers and their product

SOLUTION

Example

A piece of wire 10 feet long is divided into two portions, one being bend to form a square and the other bent to form a circle. Show that the sum of areas of the square and circle is least when the side of the square is equal to the diameter of the circle

SOLUTION

Let x be circumference of the circle and y be the perimeter of the square

Then l =10 feet

APPLICATION OF DIFFERENTIATION

The linear (or tangent line ) approximation of f(x) at x = x₀ is the function

The y – coordinate y₁ of the point m, the tangent line corresponding to x = x₁ is simply found by substituting x = x₁ into equation

we define increment dx and dy by dx =x₁ – x_oand dy = f(x₁) – f(x_o)

Using this notation, equation (3) gives the approximation

To obtain dy, subtract f (x_o) from both sides to yield

where . When using this notation, we define dx , the differential of x by dx = dx so that dy = f¹(x)dx

Example

Use a linear approximation to approximate

Solution

a). We are approximating values of the function

The closest number to 8.05 whose cube root we can easily determine is 8

The closest number to 25.4 is 27 whose cube root we know exactly as 3

Example

The radius of a circle increasing from 4cm to 4.03. Find the approximate increase in its area. Find the actual increase

Solution

Example

If the radius of a sphere decreasing by 0.5%. Find the percentage decrease in the

i) Surface Area ii) Volume

Solution

TANGENT AND NORMAL

Given the curve y= f(x), the line which touches the curve y = f(x) at point to be tangent at P is defined to the tangent at P. The gradient of tangent = dy/dx

The equation of the line at (x₁,y₁) is obtained by using y – y₁ = m(x – x₁) as

The normal point (x₁,y₁) on the curve is the line perpendicular to the tangent at that point, so its gradient is and its equation gives

Example

Find the gradient and equation of the tangent and normal to at point (1, –5)

Solution

Example

Find the equation the tangent to the curve , which is perpendicular to the point to the tangent at point (1, –1) to the curve

Solution

Example

Find the equation of the tangent to the curve which is parallel to the x – axis

Solution

RATE OF CHANGE

One of the numerous application of derivation is measuring the rate of change of function with respect to the variable

Example

A spherical ball is inflated by pumping air into it the rate of 100cm³/min. Find the rate at which the radius is increasing when the radius is 5cm

Solution

Example

Water is running out of a conical funnel at the rate of 1cm³./sec, The radius of the base if the funnel is 5cm and its is height is10cm . Find the rate at which the water level is falling when it is falling is 4cm from the top.

SOLUTION

Using similar triangles

SOLUTION

EXAMPLE 4

GENERAL THEOREM OF DIFFERENTIATION

EXAMPLE 7

EXAMPLE 8

EXAMPLE 9 Find the derivative of the following MPSetEqnAttrs('eq0014','',3,[[252,12,2,-1,-1],[335,16,3,-1,-1],[419,21,3,-1,-1],[376,18,3,-1,-1],[504,25,5,-1,-1],[630,30,6,-1,-1],[1049,50,10,-2,-2]]); MPEquation();

IMPLICIT DIFFERENTIATION

EXAMPLE 13

SOLUTION

MAXIMUM AND MINIMIUM VALUES RLATIVE EXTREMA

CONDITION FOR MAXIMA AND MINIMA

CONDITION FOR MAXIMUM AND MINIMUM POINTS

Point of Inflection

SOLUTION

EXAMPLE 44

SOLUTION

EXAMPLE 45

SOLUTION

SOLUTION

EXAMPLE 47

SOLUTION

Example 48

Example

EXAMPLE 9 Find the derivative of the following