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
| 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:
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
| dy dx | = f(x + dx) − f(x) dx | ||
| = (x + dx)2 − x2 dx | f(x) = x2 | ||
| = x2 + 2x(dx) + (dx)2 − x2 dx | Expand (x+dx)2 | ||
| = 2x(dx) + (dx)2 dx | x2−x2=0 | ||
| = 2x + dx | Simplify fraction | ||
| = 2x | dx goes towards 0 |
So the derivative of x2 is 2x
Why don't you try it on f(x) = x3 ?
| dy dx | = f(x + dx) − f(x) dx | ||
| = (x + dx)3 − x3 dx | f(x) = x3 | ||
| = x3 + ... (your turn!) dx | Expand (x+dx)3 |
What derivative do you get?