Method: Limit Definition of the Derivative

Applying the limit definition of the derivative is really just an algebra process that ends in a limit problem. This is the long way of finding a derivative, $f^{\prime }\mathrm{(}x\mathrm{)}$. Once you are past this point (homework, quiz, exam) in your calculus course, you will not really use this method. In the next sections of this course, you will learn the quicker methods to finding a derivative. Don’t skip this section though, as you will definitely have to do this on at least one calculus exam in your actual class.

Step 1: Know the Limit Definition of the Derivative.

You will usually not be given this formula, and I highly recommend you write it down (dump it out of your brain) as soon as you are able to start working on an exam. This process is not difficult if you know the formula, if you don’t the process is far more difficult.

$f^{\prime }\mathrm{(}x\mathrm{)}=\underset{h\to 0}{\mathit{lim}}\frac{f\mathrm{(}x+h\mathrm{)}–f\mathrm{(}x\mathrm{)}}{h}$

You will break this larger formula down piece by piece to make it more manageable.

Step 2: Find the  $f\mathrm{(}x\mathrm{)}$ piece.

This piece is normally just given to you as the starting equation.

Step 3: Find the $f\mathrm{(}x+h\mathrm{)}$ piece.

Apply the function notation process which is telling you to plugin $x+h$ everywhere you see an x in the original equation, $f\mathrm{(}x\mathrm{)}$. You will need to expand this portion of the equation.

Step 4: Plug the  $f\mathrm{(}x\mathrm{)}$piece and the$f\mathrm{(}x+h\mathrm{)}$piece into the limit definition of the derivative.

Always put the $f\mathrm{(}x\mathrm{)}$ piece in a set of parentheses behind minus sign. You are subtracting the entire $f\mathrm{(}x\mathrm{)}$ piece.

Step 5: Simplify the equation as much as possible, and then treat it like any other limit problem and run the limit process.