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# Basic differentiation

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## Basic differentiation

**The simplest rule of differentiation is as follows:**

**Example:**

Differentiate y = x^{3}.

**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 x^{3} (when differentiating with respect to x) is 3x^{2}.

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: **

- y = 3x
- y = sin x
- y = cos x
- y = 4x
^{3}

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 = 5x^{2} + 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 5x^{2} 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:**

- y = 10x
^{2}- 4x - y = 4sin x + 5x
- y = tan x + 8x
^{2}

**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.