Introduction to Derivatives
It is all about slope!
Slope = Change in YChange in X
We can find an average slope between two points.
But how do we find the slope at a point?
There is nothing to measure!
But with derivatives we use a small difference ...
... then have it shrink towards zero.
Let us Find a Derivative!
To find the derivative of a function y = f(x) we use the slope formula:
Slope = Change in Y Change in X = ΔyΔx
And (from diagram) we see that:
|x changes from||x||to||x+Δx|
|y changes from||f(x)||to||f(x+Δx)|
Now follow these steps:
- Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
- Simplify it as best we can
- Then make Δx shrink towards zero.
Example: the function f(x) = x2
We know f(x) = x2, and can calculate f(x+Δx) :
|Start with:||f(x+Δx) = (x+Δx)2|
|Expand (x + Δx)2:||f(x+Δx) = x2 + 2x Δx + (Δx)2|
|The slope formula is:||f(x+Δx) − f(x) Δx|
|Put in f(x+Δx) and f(x):||x2 + 2x Δx + (Δx)2 − x2 Δx|
|Simplify (x2 and −x2 cancel):||2x Δx + (Δx)2 Δx|
|Simplify more (divide through by Δx):||= 2x + Δx|
|And then as Δx heads towards 0 we get:||= 2x|
Result: the derivative of x2 is 2x
We write dx instead of "Δx heads towards 0", so "the derivative of" is commonly written
x2 = 2x
"The derivative of x2 equals 2x"
or simply "d dx of x2 equals 2x"
What does x2 = 2x mean?
It means that, for the function x2, the slope or "rate of change" at any point is 2x.
So when x=2 the slope is 2x = 4, as shown here:
Or when x=5 the slope is 2x = 10, and so on.
Note: sometimes f’(x) is also used for "the derivative of":
f’(x) = 2x
"The derivative of f(x) equals 2x"
or simply "f-dash of x equals 2x"
Let's try another example.
Example: What is x3 ?
We know f(x) = x3, and can calculate f(x+Δx) :
|Start with:||f(x+Δx) = (x+Δx)3|
|Expand (x + Δx)3:||f(x+Δx) = x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3|
|The slope formula:||f(x+Δx) − f(x) Δx|
|Put in f(x+Δx) and f(x):||x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3 − x3 Δx|
|Simplify (x3 and −x3 cancel):||3x2 Δx + 3x (Δx)2 + (Δx)3 Δx|
|Simplify more (divide through by Δx):||= 3x2 + 3x Δx + (Δx)2|
|And then as Δx heads towards 0 we get:||x3 = 3x2|
Have a play with it using the Derivative Plotter.
Derivatives of Other Functions
We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).
But in practice the usual way to find derivatives is to use:
Example: what is the derivative of sin(x) ?
On Derivative Rules it is listed as being cos(x)
Using the rules can be tricky!
Example: what is the derivative of cos(x)sin(x) ?
You can't just find the derivative of cos(x) and multiply it by the derivative of sin(x) ... you must use the "Product Rule" as explained on the Derivative Rules page.
It actually works out to be cos2(x) - sin2(x)
So that is your next step: learn how to use the rules.
"Shrink towards zero" is actually written as a limit like this:
"The derivative of f equals the limit as Δx goes to zero of f(x+Δx) - f(x) over Δx"
Or sometimes the derivative is written like this (explained on Derivatives as dy/dx):
The process of finding a derivative is called "differentiation".
You do differentiation ... to get a derivative.
Where to Next?
Go and learn how to find derivatives using Derivative Rules, and get plenty of practice.