# Derivatives as dy/dx

Derivatives are all about **change** ...

**rate of change**) at any point.

In Introduction to Derivatives *(please read it first!)* we looked at how to do a derivative using **differences** and **limits**.

Here we look at doing the same thing but using the "dy/dx" notation (also called *Leibniz's notation*) instead of limits.

We start by calling the function "y":

y = f(x)

## 1. Add Δx

When x increases by Δx, then y increases by Δy :

y + Δy = f(x + Δx)

## 2. Subtract the Two Formulas

From: | y + Δy = f(x + Δx) | |

Subtract: | y = f(x) | |

To Get: | y + Δy − y = f(x + Δx) − f(x) | |

Simplify: | Δy = f(x + Δx) − f(x) |

## 3. Rate of Change

To work out how fast (called the **rate of change**) we **divide by Δx**:

\frac{Δy}{Δx} = \frac{f(x + Δx) − f(x)}{Δx}

## 4. Reduce Δx close to 0

We can't let Δx become 0 (because that would be dividing by 0), but we can make it **head towards zero** and call it "dx":

Δx dx

You can also think of "dx" as being **infinitesimal**, or infinitely small.

Likewise Δy becomes very small and we call it "dy", to give us:

\frac{dy}{dx} = \frac{f(x + dx) − f(x)}{dx}

## Try It On A Function

Let's try f(x) = x^{2}

\frac{dy}{dx} | = \frac{f(x + dx) − f(x)}{dx} | ||

= \frac{(x + dx)^{2} − x^{2}}{dx} | f(x) = x^{2} |
||

= \frac{x^{2} + 2x(dx) + (dx)^{2} − x^{2}}{dx} | Expand (x+dx)^{2} |
||

= \frac{2x(dx) + (dx)^{2}}{dx} | Simplify (x^{2}-x^{2}=0) |
||

= 2x + dx | Simplify fraction | ||

= 2x | dx goes towards 0 |

So the derivative of **x ^{2}** is

**2x**

### Why don't you try it on f(x) = x^{3} ?

\frac{dy}{dx} | = \frac{f(x + dx) − f(x)}{dx} | ||

= \frac{(x + dx)^{3} − x^{3}}{dx} | f(x) = x^{3} |
||

= \frac{x^{3} + ... (your turn!)^{}}{dx} | Expand (x+dx)^{3} |

What derivative do *you* get?