Derivatives as dy/dx

Derivatives are all about change ...

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

3. Rate of Change

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

ΔyΔx = 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:

dy dx = f(x + dx) − f(x) dx

Try It On A Function

Let's try f(x) = x²

So the derivative of x² is 2x