❤︎² Derivatives with the Limit Definition.. How? (mathbff)

Finding derivatives using the limit defintion (also called the limit process), isn't usually the fastest or easiest way to find a derivative, but you'll probably have to learn it anyway. (And understanding it will help you understand derivatives, which will make it easier to learn the faster ways..)

To find a derivative with the limit definition, the basic strategy is always the same: Start with the limit definition and rewrite it with the function you're trying to find the derivative for. The basic structure will always be: (the function with 'X + H') minus (the function with 'X') all divided by H.

The next step to simplify and reduce until you can use direct substitution to find the limit as H approaches 0. How you do that will depend on your function, but there are some strategies that can help with different types of functions:

1) Polynomial Functions (1:26)

For polynomial functions, be aware that you may end up with a lot of terms, but the strategy is to use the same algebra techniques you already know to simplify and reduce. Often you'll be able to factor out an H in the numerator, use it to cancel the H in the denominator, and then just plug 0 in for H to find the limit.

2) Rational Functions (4:58)

For rational functions you'll probably want to combine any fractions by finding their common denominator. You'll often need to multiply by the reciprocal (1/H or similar) to move past having a single H in the denominator.

3) Radical Functions (8:44)

For radical functions a good trick to remember is that you can multiply any terms by their conjugate to remove the radicals. Removing the radicals may make it easier to factor H out.