Introduction

Before we get into differentiation questions and answers, some background (yes, you may skip past this intro!). In the 17th century Mathematicians were working on a range of problems and attempting to solve unanswered questions – as Mathematicians do! One notable problem was working with tangents to curves; the so-called ‘tangent line problem’. This enquiry, building on the work of their predecessors, eventually led to the concept of a derivative and the process of differentiation. Differentiation would prove to be of huge importance in a range of fields and formed one half of what we now call CALCULUS – the other half being integration. The history of the Calculus is pretty interesting – if you like that sort of thing. If you do, you can read more about THE HISTORY OF CALCULUS.

Differentiation Meaning

Back to the present. Differentiation is simply the process of finding the derivative of a function. When I say ‘simply’ what I really mean is ‘make me want to cry difficult!’ The definition of a derivative may be simple, but finding derivatives isn’t always so straight-forward. Firstly, you have to learn the rules (fortunately there aren’t very many but they’re easy to get muddled). Secondly, you have to know when to use those rules (i.e. not get them muddled). And finally, you’ll want to know what you can do with these rules (i.e. what can I use derivatives for?).

Differentiation Rules

That’s right, differentiation does rule! In fact, arguably it does since so much of the modern world would be impossible without differentiation. In my experience of working with many many . . . many many differentiation students, I’ve found that most are competent with the rules themselves. It’s when they’re presented in weird ways or have to be combined . . . or sometimes one rule seems to masquerade as another . . . that’s when the trouble starts! We’ll look at some examples of this later. But let’s start with the rules themselves. You have to know these or the rest is irrelevant. See next section . . .

The Rules for Differentiation

There are arguably a bunch of differentiation rules but for the vast majority of cases – and certainly when you first start with differentiation – you’ll just need to know The Power Rule, The Product Rule, The Quotient Rule, and The Chain Rule. There are also rules for Trigonometric functions, Logarithmic functions and Exponential functions but these are quite easily memorised (here’s a downloadable PDF DIFFERENTIATION FORMULA SHEET) and are, in large part, examples of The Chain Rule. Let’s look at each rule in turn.

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The Power Rule for Differentiation

The Power Rule is without doubt the first rule you should learn to be successful in differentiation. The Power Rule is for differentiating polynomial type functions. Check out our post about THE POWER RULE or watch our THE POWER RULE VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these POWER RULE QUESTIONS and POWER RULE SOLUTIONS PDF. Free, no sign-up required.

The Product Rule for Differentiation

The Product Rule is for differentiating functions which are the product of two other functions and can be a real time-saver. And often it’s the only rule that can be used. Check out our post about THE PRODUCT RULE or watch our THE PRODUCT RULE VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these PRODUCT RULE QUESTIONS and PRODUCT RULE SOLUTIONS PDF. Free, no sign-up required.

The Quotient Rule for Differentiation

The Quotient Rule is for differentiating functions which can be written as one function divided by another (i.e. a quotient). It’s usually easy to recognise when to use. Check out our post about THE QUOTIENT RULE or watch our THE QUOTIENT RULE VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these QUOTIENT RULE QUESTIONS and QUOTIENT RULE SOLUTIONS PDF. Free, no sign-up required.

The Chain Rule for Differentiation

The Chain Rule is for differentiating composite functions. It’s a powerful rule and can be used in many different scenarios but it’s not always easy to recognise when to use it. Check out our post about THE CHAIN RULE or watch our THE CHAIN RULE VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these CHAIN RULE QUESTIONS and CHAIN RULE SOLUTIONS PDF. Free, no sign-up required.

Examples of The Chain Rule for Differentiation

Since The Chain Rule can manifest is so many different ways it can be difficult for students to know when to use it. These typical examples should help you out. Check out our post about THE CHAIN RULE EXAMPLES or watch our THE CHAIN RULE EXAMPLES VIDEO to start learning.

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Applications of Differentiation

Differentiation is used across various disciplines such as science, engineering and economics. Differentiation, sometimes called ‘rate of change’ essentially measures how quickly one variable changes with respect to another variable (for example the rate of change of distance with respect to time is called velocity, which we commonly call speed). In the real world there are countless variables so you can imagine how applicable differentiation could be. But differentiation also helps us solve many Math problems and these are often called applications of derivatives. Once students learn the derivative rules they are then expected to apply them in various ways. Check out some common Math applications of derivatives below.

The Tangent Line to a Curve

The grandaddy of differentiation applications! As mentioned in the introduction this is thee classic Calculus problem and a very common question for differentiation students to have to tackle on tests. But want to know a secret? It’s actually a really easy problem type . . . well, now that we have differentiation. Of course, for those guys in the 17th century is wasn’t so much fun! Check out our post about FINDING THE TANGENT LINE TO A CURVE or watch our FINDING THE TANGENT LINE TO A CURVE VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these TANGENT LINE QUESTIONS and TANGENT LINE SOLUTIONS PDF. Free, no sign-up required.

Increasing Functions

A function can be increasing (or decreasing) either at a single point or on a range of values. Put simply, a function is increasing if the values of the function (i.e. the Y or f(x) values) are increasing. This means that the graph of the function is sloping ‘up to the right’ at the point of interest or on the range of values of interest. Determining whether a function is increasing or decreasing is a key application of derivatives. Check out our post about INCREASING FUNCTIONS or watch our INCREASING FUNCTIONS VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these INCREASING FUNCTIONS QUESTIONS and INCREASING FUNCTIONS SOLUTIONS PDF. Free, no sign-up required.

Stationary Points

I regularly see students overcomplicate and get confused by Stationary Points. If you’re one of those students then, don’t worry, this video will sort you out. In fact, the process of finding stationary points is, in itself, quite straight-forward. The challenge occurs when you’re asked to work with unusual functions which can easily throw you off. Also, determining the nature of a Stationary Point is often way overcomplicated! Check out our post about STATIONARY POINTS or watch our STATIONARY POINTS VIDEO to start learning.

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Maximum & Minimum Function Values (Local Extrema)

Finding the maximum or minimum values of a function on an interval (also called finding local extrema) is a very common differentiation question, and one that students should expect to see on assessments. Although it sounds quite complicated the technique is actually straight-forward and really an application of Stationary Points – hopefully you already know about those?! If not, see our STATIONARY POINTS VIDEO. Check out our post about LOCAL EXTREMA or watch our LOCAL EXTREMA VIDEO to start learning.

Got the idea? Prove it by checking out our differentiation questions and answers. Check your knowledge with these LOCAL EXTREMA QUESTIONS and LOCAL EXTREMA SOLUTIONS PDF. Free, no sign-up required.