Basic differentiation

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Basic differentiation

The simplest rule of differentiation is as follows:

Simplest rule


Differentiate y = x3.


(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:

Rules Rules

(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:

Addition and subtraction

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:

Addition and subtraction

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.

S-cool exclusive!!