Introduction to Derivatives
It is all about slope!
Slope = 

You can find an average slope between two points.  
But how do you find the slope at a point? There is nothing to measure! 

But with derivatives you use a small difference ... ... then have it shrink towards zero. 
Let us Find a Derivative!
We will use the slope formula:
Slope =  Change in Y  =  Δy 
Change in X  Δx 
To find the derivative of a function y = f(x)
And follow these steps:
· Fill in this slope formula: 


· Simplify it as best you can,  
· Then make Δx shrink towards zero. 
Like this:
Example: the function f(x) = x^{2}
We know f(x) = x^{2}, and can calculate f(x+Δx) :
Start with:  f(x+Δx) = (x+Δx)^{2}  
Expand (x + Δx)^{2}:  f(x+Δx) = x^{2} + 2x Δx + (Δx)^{2} 
Fill in the slope formula: 


Simplify (x^{2} and^{} x^{2} cancel): 


Simplify more (divide through by Δx):  = 2x + Δx  
And then as Δx heads towards 0 we get:  = 2x 
Result: the derivative of x^{2} is 2x
And we write dx instead of "Δx heads towards 0", so the common notation for "the derivative of" is :
x^{2} = 2x
"The derivative of x^{2} equals 2x"
or simply "d dx of x^{2} equals 2x"
Sometimes f’(x) is also used for "the derivative of":
f’(x) = 2x
"The derivative of f(x) equals 2x"
What does x^{2} = 2x mean?It means that, for the function x^{2}, the slope or "rate of change" at any point is 2x. So, when x=2, the slope will be 2x = 4, as shown here: Or when x=5 the slope will be 2x = 10, and so on. 
Let's try another example.
Example: What is x^{3} ?
We know f(x) = x^{3}, and can calculate f(x+Δx) :
Start with:  f(x+Δx) = (x+Δx)^{3}  
Expand (x + Δx)^{3}:  f(x+Δx) = x^{3} + 3x^{2} Δx + 3x (Δx)^{2} + (Δx)^{3} 
Put it in the slope formula: 


Simplify (x^{3} and^{} x^{3} cancel): 


Simplify more (divide through by Δx):  = 3x^{2} + 3x Δx + (Δx)^{2}  
And then as Δx heads towards 0 we get:  x^{3} = 3x^{2} 
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:
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 cos^{2}(x)  sin^{2}(x)
So that is your next step: learn how to use the rules.
Notation
"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.