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) |
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Subtract: |
y = f(x) |
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To Get: |
y + Δy - y = f(x + Δx) - f(x) |
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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) = x2
f(x) = x2 | |||
Expand (x+dx)2 | |||
Simplify (x2−x2=0) | |||
Simplify fraction | |||
dx goes (infinitely small) |