general form of the derivative using the limit
But maybe the delta x's bother you, so we'll use h's instead. (This is the one that's illustrated in the picture above.)
Alternative general form of the derivative using the limit
The derivative at a point x = c
So how and why does it work?
This is a great website that explains the (pardon the pun) derivation of the limit definition of the derivative:
Developing the Limit Definition of the Derivative
Here's another site with a helpful explanation:
Paul's Online Notes: Limit Definition of the Derivative
Developing the Limit Definition of the Derivative
Here's another site with a helpful explanation:
Paul's Online Notes: Limit Definition of the Derivative
Here's a 15-minute YouTube video that you might find helpful:
If you'd like some additional instructional YouTube videos, Khan Academy has three good ones also. They're called: "Calculus: Derivatives 1", "Calculus: Derivatives 2", and "Calculus: Derivatives 2.5".
Ready for practice?
You should probably have some paper handy. This website provides 12 problems, in which you should practice using the limit definition to find the derivative. (Use one of the first two forms listed at the top of the page, since you'll be finding the general derivative.) You'll be able to check your answers when you finish each problem.
Practice Problems
Practice Problems