Chapters

## What Is Differentiation?

As defined by S-cool.co.uk, Differentiation is 'a tool of mathematics that is primarily used for **calculating rates of change**'. The tool thus allows us to find the **rate of change of velocity with respect to time** (i.e. acceleration) as well as the **rate of change of x with respect to y on a graph **(i.e. the gradient of the curve).

As part of your A Level course, your teacher will no doubt teach you **the basic rules of Differentiation**, including Differentiating *x to the power of something*, along with **studying Notation**. Finally, you will use all of this knowledge to Differentiate the equation of a curve, **finding a formula **for its gradient.

If all of this sounds a bit perplexing, rest assured that your Maths teacher will be able to explain Maths processes and functions so that they come to you easily. Until then, why not take a look at our guide below which sets out some of the basic Differentiation rules, as well as a few more complicated ones for those seeking more of a challenge!

## When And Where Might I Need To Use Differentiation?

Historically, Differentiation was used by seamen to **better understand how the Earth, stars and planets in the solar system move with respect to one another**. However, putting manual navigation at sea to one side for now, what are the **other uses for Differentiation** in modern society?

Differentiation and integration are tools that can help us to solve many **real-world problems**. For example, various industries make use of derivatives to **find out the minimum and maximum values of things**, like cost, profit, loss, strength and the quantity of material used in a buildings object.

Within the world of industry, this term is referred to

optimisation.

The subject also crops up quite regularly for those **working in the field of science and engineering**, particularly when looking at the **behaviour and trends of moving objects**.

However, even those who are not working directly with Maths, Science, Computer Science or Physics will still use some of these tools without even knowing it.

For instance, each time somebody in business uses a trend to **anticipate something happening in the future**, or thinks about **how fast their return will multiply over a specified time**, they are using optimisation methods, which as we now know are directly related to Differentiation and Integration.

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That does not mean, however, that many people can do this well. This is where an A Level and/or degree in Maths comes in. By studying Maths at this advanced level, you will learn **how to do the above techniques correctly and consistently** so that you can take advantage of optimisation tools in the future.

## An Introduction To Basic Differentiation Techniques

In Calculus, when you have an equation for *y* written in terms of *x*, it's easy to use **basic differentiation techniques to find the derivative**. These are known to mathematicians as **Explicit Differentiation** techniques.

However, for equations that are more complicated to rearrange, for instance when *y* is by itself on one side of the *=* sign, a different approach is needed. The method used to **work out multi-variable equations** is **Implicit Differentiation**, which is easy if you already know how to use Explicit Differentiation!

### The Basic Rules Of Simple Differentiation

#### The Constant Rule

While some of you might be wondering if there is such a thing as basic Differentiation, there really is! Take the **constant rule**, for example, which is as follows

\[f (x) = 5\] is a line with a gradient of *0*, therefore it’s derivative is also *0*

As such, if \[f (x) = c (any number)\] then \[f’(x) = 0\]

#### The Power Rule

If [latexpage] \[f(x) = x^4\] then to find its derivative you need to **take the power and bring it in front of the x **before reducing the power by

*1*. Here, this would mean that \[f’(x) = 4x^3\] Once you have understood this simple process, repeat it again and again. This rule, like the one above, is constant and works for any power, whether it is

**positive, negative or a fraction**.

Note: \[x to the zero power = 1\]

#### The Constant Multiple Rule

Even if the function you are setting out to differentiate begins with **a coefficient**, this has no effect on the process you adopt. Go ahead and differentiate the function **using the appropriate rule**, with the coefficient staying in place until the final step when you simplify the answer by multiplying by the presented coefficient. So:

If you see \[y = 5x^3\] you can deduce that the derivative of \[x^3\] is \[3x^2\] and therefore that the derivative of \[5(x^3)\] is \[5(3x^2)\] It is only at the end that you simplify in this way: \[5(3x^2)\] equals \[15x^2\] so \[y’ = 15x^2\]

#### The Sum Rule

When faced with a sum of terms, you can tackle the problem by working out the derivative of **each term on its own**, like this.

Question

What is \[f’(x)\] if \[f(x) = x^6 + x^3 + x^2+ x + 10\]

Answer

\[f’(x) = 6x^5+ 3x^2 + 2x + 1\]

#### The Difference Rule

The opposite of a sum, a difference acts just like a subtraction. You would therefore solve as follows, leaving the subtraction sign unchanged.

Problem

If \[y =3x^5 – x^4 – 2x^3 + 6x^2+ 5x\] then

Answer

\[y’= 15x^4 – 4x^3 – 6x^2 + 12x + 5\]

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## An Introduction To Integration Techniques

When it comes to the term Integration, this covers yet more subtopics and techniques. The principal topics within this chapter are **Integration by Parts, Trig Substitutions, Integrals Involving Trig Functions, Integrals Involving Roots, Integrals Involving Quadratics, Partial Fractions, Integration Strategy, Approximating Definite Integrals, Improper Integrals and, finally, Comparison Tests for Improper Integrals**.

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### A First Taste Of Integration

With so many different techniques to explore, let's focus on Integration by Parts, which is the area in which the most pupils will come across **during the course of their Maths studies**.

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### Integration by Parts

Integration by Parts is the special mathematical process that you adopt to

integrate products of two functions.

Functions sometimes crop up as **products of other functions**, and may need us to integrate them. A rule exists for this process and, once grasped, can be use to **derive all relevant Integration by Parts problems**.

The technique will enable you to state the formula for Integration by Parts and the integrate the products of functions using the technique. Still none the wiser? Take a look at these examples.

When approaching an Integration by Parts problem, the main thing to remember is to **rearrange the formula**.

For instance, we know that if

\[y = uv\] then \[\frac{\text{d}y}{\text{d}x} = \frac{\text{d}(uv)}{\text{d}x} = u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}\]

Once you have rearranged the formula, you can then safely integrate on both sides like

\[`\int_{}^{} u \frac{\text{d}v}{\text{d}x} dx = \int_{}^{} \frac{\text{d}(uv)}{\text{d}x} dx - \int_{}^{} v \frac{\text{d}u}{\text{d}x}dx`\]

before the final step, which is to simplify the entire formula.This will ultimately leave you with

\[`\int_{}^{} u \frac{\text{d}v}{\text{d}x} dx = u v - \int_{}^{} v \frac{\text{d}u}{\text{d}x}dx`\]

If you are a English student too, think of the above process of simplifying the formula as sort of

a way to write it in shorthand.

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### Why Do I Need To Learn Integration Techniques?

When we have so many **pieces of software** that can now use Integration techniques to simplify and work out formulas, why do we need to learn about the methods as students?

The simple answer is that nobody really knows if they will actually use these tools in the future, but isn't it nice to **know how the technology you are using operates** and to know that you could potentially do the working out in the event that computers ever become extinct, albeit a little slower (not that we can expect that to happen any time soon!)?

While some may think that these techniques have **no practical application** and are therefore a bit of a waste of time, it is important to remind oneself that **the professionals behind the syllabus** are merely trying to help you **build up an intellectual understanding of complicated mathematical processes **while encouraging you to **develop logical intuition**.

Remember also that, while your A Level course may offer a somewhat rigid **approach to learning techniques related to Differentiation and Integration**, you will have the opportunity in future to learn different patterns, some that may even be **self-taught**, if you ever decide to follows steps and processes **without the aid of a computer**. Or you can learn them with maths tutors who will help you find the technique best suited to your learning style.

Whether this is as **part of a degree, a Masters degree or during your mathematical career**, you will no doubt find some (if not tonnes of) uses for **the methods taught during this A Level course**.

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