# Introduction to Derivatives

It is all about slope!

Slope = \frac{Change in Y}{Change in X} |

We can find an
| ||

But how do we find the slope There is nothing to measure! | ||

But with derivatives we use a small difference ... ... then have it |

## Let us Find a Derivative!

To find the derivative of a function y = f(x) we use the slope formula:

Slope = \frac{Change in Y}{Change in X} = \frac{Δy}{Δx}

And (from the diagram) we see that:

x changes from | x | to | x+Δx | |

y changes from | f(x) | to | f(x+Δx) |

Now follow these steps:

- Fill in this slope formula: \frac{Δy}{Δx} = \frac{f(x+Δx) − f(x)}{Δx}
- Simplify it as best we can
- Then make
**Δx**shrink towards zero.

Like this:

### Example: the function **f(x) = x**^{2}

^{2}

We know **f(x) = x ^{2}**, and we can calculate

**f(x+Δx)**:

Start with: | f(x+Δx) = (x+Δx)^{2} | |

Expand (x + Δx)^{2}: | f(x+Δx) = x^{2} + 2x Δx + (Δx)^{2} |

**f(x+Δx)**and

**f(x)**:\frac{x^{2} + 2x Δx + (Δx)^{2} − x^{2}}{Δx}

^{2}and −x

^{2}cancel):\frac{2x Δx + (Δx)^{2}}{Δx}

**as Δx**

**heads towards 0**we get:= 2x

Result: the derivative of ** x ^{2}** is

**2x**

In other words, the slope at x is **2x**

We write **dx** instead of **"Δx**** heads towards 0"**.

And "the derivative of" is commonly written :

x^{2} = 2x*"The derivative of x^{2} equals 2x"*

or simply

*"d dx of*

**x**equals^{2}**2x**"### What does x^{2} = 2x mean?

It means that, for the function x^{2}, the slope or "rate of change" at any point is **2x**.

So when **x=2** the slope is **2x = 4**, as shown here:

Or when **x=5** the slope is **2x = 10**, and so on.

Note: sometimes f’(x) is also used for "the derivative of":

f’(x) = 2x*"The derivative of f(x) equals 2x"*

or simply *"f-dash of x equals 2x" *

Let's try another example.

### Example: What is x^{3} ?

We know **f(x) = x ^{3}**, and can calculate

**f(x+Δx)**:

Start with: | f(x+Δx) = (x+Δx)^{3} | |

Expand (x + Δx)^{3}: | f(x+Δx) = x^{3} + 3x^{2} Δx + 3x (Δx)^{2} + (Δx)^{3} |

**f(x+Δx)**and

**f(x)**:\frac{x^{3} + 3x^{2} Δx + 3x (Δx)^{2} + (Δx)^{3} − x^{3}}{Δx}

^{3}and −x

^{3}cancel):\frac{3x^{2} Δx + 3x (Δx)^{2} + (Δx)^{3}}{Δx}

^{2}+ 3x Δx + (Δx)

^{2}

**as Δx**

**heads towards 0**we get:= 3x

^{2}

Result: the derivative of ** x ^{3}** is

**3x**

^{2}Have a play with it using the Derivative Plotter.

## Derivatives of Other Functions

We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).

But **in practice** the usual way to find derivatives is to use:

Using the rules can be tricky!

### Example: what is the derivative of cos(x)sin(x) ?

You can't just find the derivative of cos(x) and multiply it by the derivative of sin(x) ... you must use the "Product Rule" as explained on the Derivative Rules page.

It actually works out to be **cos ^{2}(x) − sin^{2}(x)**

So that is your next step: learn how to use the rules.

## Notation

"Shrink towards zero" is actually written as a limit like this:

"The derivative of **f** equals **the limit as Δx goes to zero** of f(x+Δx) - f(x) over Δx"

Or sometimes the derivative is written like this (explained on Derivatives as dy/dx):

The process of finding a derivative is called "differentiation".

You **do** differentiation ... to **get** a derivative.

## Where to Next?

Go and learn how to find derivatives using Derivative Rules, and get plenty of practice: