If you think secondary school maths is difficult, wait till you begin maths A level and if you already have, you will know it is a whole different breed of maths.
People argue that maths is easy, it is true in some instances and not true in some, what we know is with the right preparation you can solve maths in whatever form it appears before you. Normal maths, advanced level maths, etc.
Absolute devotion to learning and studying it, and solving its equations and problems is one way to go about it, burning the midnight candle as it were, that works too, but when we said preparations we were thinking of something more efficient, such as getting a maths tutor to help you solve math problems.
A maths tutor will not only prove useful to your knowledge but they will add to your confidence at school, because now you have someone, someplace that you can go to, to have all those questions answered, and those things you don’t understand explained.
Tutors have the experience and have already done all the mistakes you will likely make, so if you have a maths tutor you can avoid those mistakes when you sit to solve A level maths questions.
You can also prepare for and solve maths by practicing with A level maths past papers, and by finding the maths A level curriculum beforehand.
Just as algebraic functions and a few other topics dominated secondary school maths, there are some fundamental topics that make up A level maths. Some are extensions of secondary school maths like trigonometry, while some are totally brand new, like calculus.
Other maths A level topics include; Calculus, Sequences and series, Trigonometry, Exponentials and logarithms, Differentiation, Integration, Numerical methods, Vectors.
In this post, we will be looking at a branch of mathematics known as calculus. It is of two types, differential calculus where you use differentiation to solve maths problems, and integral calculus wherein you use integration to solve maths problems.
What Is Differentiation?
Differentiation is a mathematical principle we use to solve the rate of change. It is used to calculate the rate of change of physical quantities such as speed and force with respect to time.
And also the rate of change of 'x' with respect to 'y' on a graph, better known as the gradient of a curve. Differentiation enables us to pinpoint any certain point on a curve in a graph, and calculate it’s value.
Differentiation is solved in many forms, including differentiating x to the power of a number, and by studying notation. But most excitingly, differentiation is primarily used to find a formula for the gradient of a curve, by differentiating the equation of the curve.
So that any value as long as it is on the curve can be calculated at any moment in time.
Your A level maths teacher is sure to explain to you everything you need to know about differentiating equations and gradients of curves, in-depth. If you are targeting a career in the engineering or tech industry, then your best bet is to go to a university that teaches sciences. They offer A level maths and teach it distinctively well.
Universities such as FUTMINNA, ABU Zaria and Ondo State University of Science and Technology, are focused on providing the best technical knowledge possible in Nigeria. So why don’t you check them out if you want to apply, and other similar schools.
Talking about careers, let’s see if differentiation is of any use to you in the real world anyway, or if it’s just a set of numbers and letters jumbled together?
Is Differentiation Useful To Me?
It is popularly joked that all the equations and problems in maths will not amount to anything in your life, memes say “when will I ever use this” along with a picture of some diabolical maths solution.
But in truth, maths is everywhere, it is in everything we do. From the basic counting of money and hens in our farm to the more advanced calculations in maths like differentiation that lets us know the capacity of a dam or the number of materials used in a building.
That’s right, you can actually calculate and find the maximum and minimum value of almost anything, such as the number of materials used in a building we just mentioned, or the overall amount of water a manufactured tank can take, or even profit and loss.
You can use differentiation to tell how high you should stack a load without damaging the load at the bottom.
If you are going anywhere near engineering you should know that differentiation will become your cup of tea, you will be using it to solve all sorts of number-related problems. But if you are not going for any technical career you may see less of differentiation and integration.
Unless of course, you are in the financial sector. They make great use of differentiation to predict and evaluate the cost of things and to calculate the profitability of a venture.
So, as you can see maths A level's differentiation is something vital to our economy and infrastructure. It helps us solve a lot of maths problems and also helps us accurately predict the max of what we can achieve across various industries, thereby saving time and effort.
It has been used since antiquity to solve for the position of the stars and moon to ensure safe sea travel and so on. Differentiation is useful to us in so many ways we cannot begin to count, but we can only use it as far as our knowledge of it stretches.
That is why a degree in maths or a diploma in A level maths is important to learn differentiation in Nigeria, it is crucial to your surviving the next stage of life, namely the working or professional stage.
That way you will have a skillset worthy of being chosen in job interviews and one useful to you and everyone else.
Introduction To Differentiation Techniques
Differentiation is governed by some specific rules, they are the rules of differentiation, they outline how to solve differentiation questions when they appear to have certain properties such as when a value or number comes with an index, a ‘raise to power'.
Or when they appear to have a fixed form. These rules come into play and differentiation is solved using them. The rules act as guidelines and formulas for solving differentiation.
Many calculus problems and questions can be solved using a technique known as Explicit Differentiation Technique. It involves finding the derivative if y as part of an equation is written in terms of x.
But when you have more complex equations it becomes a bit more challenging, a different approach is used to calculate for the values of x and y. For instance when y is on the other side of the equal sign, (on the left or right it doesn’t matter), then Implicit Differentiation is used to solve the equation.
Some Rules of Basic Differentiation: The Constant Rule
The constant rule states that if,
f (x) = c (number), and the gradient of such equation is zero, then the derivative is also zero,
f' (x) = 0.
This is known as the constant rule, and it is by far as anyone can tell the simplest rule in calculus.
The Power Rule
If you have a function of x that has a power, i.e. a raised to power something, then to find the derivative f' you first start by taking the power and placing it next to the x as a whole number on the left side of the equation and then lower the power on the x by one. That is subtract 1 from whatever the power is.
If it’s a 4 it becomes a 3, if it’s a 6 it becomes a 5 and so on. In written form it looks like this,
If f (x) = x5 then the derivative is f' (x) = 5x4.
Introduction To Integration In Calculus
Integration is the opposite of differentiation, it is used to calculate the spaces under curves and between them. As opposed to the gradient of the curves themselves.
And compared to differentiation it has many more techniques and methods and subtopics under it, you will find integration topics such as; Integration by Parts, Trigonometric Substitutions, Integrals Involving Trig Functions, Integrals Involving Roots, Integrals Involving Quadratics, Partial Fractions, Approximating Definite Integrals, Improper Integrals and Comparison Tests for Improper Integrals.
Note that these are not all of the topics under integration, it’s just to give you a taste and an insight. Your maths teacher will cover all the topics and related subtopics with you in your maths A level course.
Like differentiation, integration has places where its application helps us accomplish our day to day activities. You can develop a software that uses the great power of these mathematical concepts to help predict, evaluate, and infer from thousands of values and give us back accurate results.