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Derivatives with the Limit Definition.. How?

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.

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Find a Limit Algebraically.. How?

There isn’t any one algebraic method that will works for finding every limit. Here are the different methods:

1) Direct Substitution (0:22) When trying to evaluate a limit algebraically your “go to" method should be direct substitution. If direct substitution works, simply plug in the value that x is approaching, and evaluate the function.

2) What about if I get 0/0? (1:25) If you try direct substitution and get 0/0 (the indeterminate form), you should try to simplify the function and then try direct substitution again.

3) Factor to Simplify (2:17) If you see higher powered polynomials in the function, you should try a form of factoring to simplify the function. Whether it is a difference of squares or cubes, a perfect square trinomial, a trinomial with or, or even a grouping situation, factoring will usually get you to the result. Once you’ve simplified by factoring, try direct substitution again.

4) Use a Conjugate to Simplify (4:28) If you see a root in the function, you should try multiplying by the conjugate of the root to simplify the function. Once the function is simplified, try direct substitution again.

5) Use Expansion of Polynomials to Simplify (7:09) If you tried factoring and weren’t able to simplify, you can try expanding any polynomials which are raised to a power, then check for anything that cancels out. Once you’ve simplified by canceling out terms, try direct substitution again.

6) Simplify With a Common Denominator (8:38) If you have a complex fraction, you can try using a common denominator to simplify the fractions. Once you’ve simplified the fractions, try direct substitution again.

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Gaussian Elimination.. How?

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Matrix Determinants.. How?

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Matrix Multiplication.. How?