With changing time, people have changed the pattern and have initiated to fight a war in a new way. With advancing technology, war has taken a different form. Earlier, team with less armor were disrespected while, people now believe that a less powerful country can cause severe damage. Hence they revere the less powerful countries as well.

New conspiracies being produced and some of which being unveiled, people pick their arms and ammunitions to set fire to rival groups, but often happen to retreat their soldiers because of untimely agreement.

Nature of war has changed to a great extent nuclear weapons have resulted to dire consequences. Nuclear weapons, these days have become inevitable. A single missile has the ability to devastate the entire earth.

The general perceptions about war have changed greatly nowadays. Paltry groups or countries appreciate modern techniques of war and are able to adapt these changes. The change in nature and manifestations of war has sown seed for a bigger disaster in the near future.

Students have changed the pattern of their study as well. The old and outdated trend has been left far behind and people have adapted modern techniques with better chances and options to learn. At this point of time, people have utilized technology as a medium for learning. Various forms of modern technology have supported education programs like distance learning. Sitting in a different corner of the globe, a student can attend college of a different country. Dreams of past have now turned into reality, it’s not a big deal for any student to use the internet as reference. Internet has largely been used as a place where you can choose the best answer for you among thousands of available ones. People are now independent to some extent. Being concentrated on theoretical knowledge, people were found to be bookworms. They could tell about the content of any topic in the books, but they couldn’t apply the logic in their real life. The things have changed since modern learning replaced the tendency of students to be limited to a room. Students can now utilize the resources available and are able to explore the world with which minds broaden and perceptions change.

When a topic is taught with a book and with use of few logics especially in science subjects, students find it too boring and difficult to understand, and for sure, it will not be a worthy lecture. But, when the same thing is taught differently using some instruments and apparatus, students’ often find it fun oriented learning. Likewise, technological advancements have caused to uplift the status of practical learning useful for day to day activities.

Modern technology has great influence on the educational sector as it has brought admirable changes. It has let students explore a different world with better prospects.

It can be argued that modern technology makes life easier and more dignified for many people. It has improved the way of learning. It has changed our lifestyle and the way of learning too. In the past decade, technology was not reliable. People used to struggle had to achieve their goal. There were no well-equipped machines or things to do work. People struggled themselves to achieve their goal. For example, our ancestors used to write with a pen made with bamboo. They used to labor hard but now the writings no more need bamboo stick to write. Robots and machines perform on their own once given command. Use of modern technology has changed a lot of the behavior of the people and their learning style.

A lot of money on developing the weapon industry also falls in improvements of learning but it is considered as a wrong fact. The modern technology should either stop or prevent social disproportions. Similarly, modern technology has made us improve learning in many aspects. It helps for the newspaper advertising, the invention of new technology, commodities, rapid development, etc. it also teaches skills of agriculture.

We all know that the modern technology has kept on improving our lifestyle and kept us forwarding in life. It has changed a way of learning. It helps to save our civilization, our way of learning and our life style. It has taught how to be unified. We have to pay a proactive role from our individual level to improving our learning.

Positivism is an philosophy which is based on social as well as natural sciences and more in logical and mathematical treatments. This theory was founded by Auguste comte and developed by Saint simon and comte. Positivists assume that sociological explanations should be like those of natural sciences and sociologists should use logic , methods and procedures of natural science. They believe social reality is objective and it can be measured and discovered by scientific methods. They also believe that social reality can be predicted for example with the help of Durkheims work on suicide we can find out the cause why people are dying . Positivists believe that society is an base for individual for example in the suicide case until and unless society creates certain circumstances an individual wont die. So this theory sees society on macro level and on subjective basis.

Whereas Anti positivism is a philosophy which says that society is subjective. It sees society as totally different from natural sciences. It says a person has different perspective and it cannot be seen in terms of quantities. It also says that there is no casual law which can govern the social behaviour. An example of this can be Goffman’s total institution. It sees society in micr level and subjectively

Seeing both the approaches there is a grear debate between positivism and anti positivism . These are the two theories that is basic theory of sociology. Nowadays we can see that both of these theories are functioning equally in the society and it is true also that positivists say society can mak an individual  and we can find many cases. Similarly anti positivists say individual make society , Gandhiji , Hitler and many more great personalities are examples of this So this is a debate thta is going on and in future too it will be going on.

The Product Rule

The product rule is used when differentiating two functions that are being multiplied together. In some cases it will be possible to simply multiply them out.

Example:

Differentiate y = x²(x² + 2x − 3).

Here y = x⁴ + 2x³ − 3x² and so:

However functions like y = 2x(x² + 1)⁵ and y = xe^3x are either more difficult or impossible to expand and so we need a new technique.

The product rule states that for two functions, u and v. If y = uv, then:

For our example:

y = 2x(x² − 1)⁵

u = 2x

v = (x² − 1)⁵

Therefore:

After factorising:

Note: After using the product rule you will normally be able to factorise the derivative and then you can find the stationary points.

For our second example:

y = xe^3x, find the turning point and sketch the graph.

u = x

v = e^3x

Therefore:

This means there is a stationary point when x = -1/3 (e^3x ≠ 0).

Also, when x = -1/3, y = -e^-1/3 = -0.123 (3sf).

By using the second derivative, which we find by using the product rule again, we can determine whether this is a maximum or a minimum.

when x = -1/3

Therefore there is a minimum at (-1/3, -0.123)

To sketch the graph we know that:

1. When x = 0, y = 0
2. There is a minimum at (-1/3, -0.123)
3. As x → ∞, y → ∞
4. As x → −∞, y → 0 (and is negative)

Therefore the graph looks like this:

The quotient rule is actually the product rule in disguise and is used when differentiating a fraction.

The quotient rule states that for two functions, u and v,

(See if you can use the product rule and the chain rule on y = uv^-1 to derive this formula.)

Example:

Differentiate

Solution:

Using a substitution to help differentiate

We will often need to differentiate functions that are more complex than the ones that we can already do. They will simply be variations, where ‘x’ has been replaced by a function ‘f(x)’.

Example:

Differentiate the following:

1. y = (3x − 2)⁴
2. y = sin 5x
3. y = 2e^{(2x + 1)}
4. y = cos 4x

In each of these cases we can use a substitution to turn the expression into something we can differentiate.

Answer to 1:

We know how to differentiate x⁴, so we use the substitution u = (3x − 2) to turn the function into something that we can differentiate. This gives:

y = (3x − 2)⁴

Let u = 3x − 2 to give us,

y = u⁴,

Now differentiate to get:

The only problem is that we want dy/dx, not dy/du, and this is where we use the chain rule.

The chain rule says that

So all we need to do is to multiply dy/du by du/dx.

As u = 3x − 2, du/dx = 3, so

3 = 12(3x − 2)³

Answer to 2:

Differentiate y = sin 5x.

Let u = 5x (therefore, y = sin u)

so using the chain rule

So when using the chain rule:

1. Express the original function as a simpler function of u, where u is a function of x.

Differentiate the two functions you now have.

Multiply the derivatives together, leaving your answer in terms of the original question (i.e. in x’s rather than u’s).

For 3 and 4, see if you can do the workings and then check your answers against these:

Answer to 3:

Answer to 4:

When familiar with the chain rule, it is possible to produce a correct answer instantly without having to write down all the substitution working; simply follow through the three steps together.

Example:

Differentiate ln(x² + 3x + 3)

The denominator is from dy/du = 1/u, the numerator is du/dx)

In each of these formulae we have used the substitution u = f(x) and so the f ′ (x) corresponds to

Equations of Tangents and Normals

As mentioned before, the main use for differentiation is to find the gradient of a function at any point on the graph. Having found the gradient at a specific point we can use our coordinate geometry skills to find the equation of the tangent to the curve.

To do this we:

1. Differentiate the function.

2. Put in the x-value into

to find the gradient of the tangent.

3. Put in the x-value into the function (y = …) to find the coordinates of the point where the tangent touches the curve.

4. Put these values into the formula for a straight line:

y – y₁ = m (x – x₁)

where m = gradient and (x₁, y₁) is where the tangent meets the curve.

Example:

Find the equation of the tangent to the curve y = x² – 2x – 3, when x = -1:

so when x = -1,

and y = 1 + 2 – 3 = 0

Therefore the equation of the tangent is y − 0 = -4(x + 1)

So now we know that y = -4x – 4 is the equation of the tangent at (-1, 0).

The normal to a curve is the line at right angles to the curve at a particular point.

This means that the normal is perpendicular to the tangent and therefore the gradient of the normal is -1 × the gradient of the tangent.

To find the equation of the normal, follow the same procedures as before, (remembering to multiply the gradient of the tangent by -1 to calculate the gradient of the normal).

Example:

Find the equation of the normal to the curve y = x³ + x − 10 when x = 2.

and y = 8 + 2 − 10 = 0

As the gradient of the tangent = 13, the gradient of the normal = -1/13

Therefore the equation of the normal is:

y − 0 = -(x – 2) / 13

Therefore: 13y = 2 − x is the equation of the normal at (2, 0).

As mentioned before, the main use for differentiation is to find the gradient of a function at any point on the graph. In particular differentiation is useful to find one of the main features of the graph – the Stationary Points.

These are points where the gradient = 0.

There are three types of stationary point:

There are three possible ways to determine the nature of a stationary point.

1. From experience – if you know the shape of the graph, then you know if it is a max/min. All quadratics where the co-efficient of x² is positive have a minimum (∪ – shaped); all quadratics where the co-efficient of x² is negative have a maximum (∩ – shaped).

2. By looking at the gradient either side of the stationary point.

3. By using the second derivative,

which often shown as

For a particular value for x, when

An example using all three methods:

Find the coordinates and nature of the turning points of the curve y = x³ − 12x + 2

Firstly, where are the stationary points?

Find where the gradient = 0.

Therefore:

when x = 2 (and y = -14) or when x = -2 (and y = 18).

Method 1:

We know that a + x³ graph has a maximum followed by a minimum, so (-2, 18) must be a maximum, and (2, -14) must be a minimum. (Also the value of the y-coordinates confirms that this must be true.)

Method 2:

For this graph the gradient = 0 when x = -2 and x = 2.

We can use the fact that the gradient is a multiple of (x + 2)(x – 2) to determine the sign of the gradient either side of these values. This is best illustrated in a table.

At x = -2, the gradient goes from positive to negative. This is a ∩-shape, and means that there is a maximum at (-2, 18).

At x = 2, the gradient goes from negative to positive. This is a ∪-shape, and means that there is a minimum at (2, -14).

Method 3:

When x = -2,

Therefore there is a maximum at (-2, 18).

When x = 2,

Therefore there is a minimum at (2, -14).

As a guideline – if you know the graph use method 1. Otherwise use method 3 unless the second derivative is hard to find.

If you do not know the shape of the graph, or you cannot differentiate twice or

then use the method 2.

Rules of differentiation

The simplest rule of differentiation is as follows:

Example:

Differentiate y = x³.

Working:

(We can see that n = 3 and a = 1 in this example so replace n with 3 and a with 1 to get:)

Note: An alternative way of writing the workings is to say:

This is the mathematical way for saying that the derivative of x³ (when differentiating with respect to x) is 3x².

There are a number of rules that are the starting points for all the hardest work. These are shown in these 2 tables:

(These rules are all listed in the revision summary, which you can print out and keep looking at to help you remember them.)

Even if you know how to use the rules above, read the examples below as they will get you warmed up for the next question session…

Using the list of rules above, work out the derivatives of the following function. Write your answers on a sheet of paper and then click for the answers to check you have done this correctly.

Some questions for you to try

Find the differentials with respect to x of:

1. y = 3x
2. y = sin x
3. y = cos x
4. y = 4x³

So far we have learnt to differentiate simple functions, such as y = 5x.

However, we also need to know how to differentiate more complex functions such as y = 5x² + 2x + 6.

To do this we need to understand how to deal with the addition or subtraction of a number of terms.

All you have to do is use the rules you have already learnt to differentiate each component of the equation.

So first we differentiate 5x² to get 10x.

Then we differentiate 2x to get 2.

Finally, we differentiate 6 to get nil.

Then we can simply add these together to give:

Try these examples to make sure you understand:

Find the differentials with respect to x of:

1. y = 10x² – 4x
2. y = 4sin x + 5x
3. y = tan x + 8x²

The mathematical way of expressing what we have just done is this:

You differentiate each term separately.

Note: If asked to differentiate a function like, t = 4 sin u we use the same ideas but different letters to get:

This means that the derivative of t, with respect to the variable u, is 4 cos u.