# Derivatives as dy/dx

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

... they show how fast something is changing (called the

**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**:

## 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:

## Try It On A Function

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

f(x) = x^{2} |
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Expand (x+dx)^{2} |
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Simplify (x^{2}-x^{2}=0) |
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Simplify fraction | |||

dx goes towards 0 |