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
If 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 very small, and call it "dx":
Δx dx
You can 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}  
Expand (x+dx)^{2}  
Simplify (x^{2}−x^{2}=0)  
Simplify fraction  
dx goes (infinitely small) 